Appendix C - Vectors and Linear Algebra

Introduction

This appendix introduces some aspects of linear algebra and complex algebra that will be helpful for the course. In addition, Dirac notation is introduced and explained.

Vectors

Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics.

Real Vectors

The simple definition of a vector --- an object that has magnitude and direction --- is helpful to keep in mind even when dealing with complex and/or abstract vectors as we will here. In three dimensional space, a vector is often written as

${\vec {v}}=v_{x}{\hat {x}}+v_{y}{\hat {y}}+v_{z}{\hat {z}},$

where the hat (${\hat {\cdot }}\,\!$) denotes a unit vector and the components $v_{i}\,\!$, $i=x,y,z\,\!$ are just numbers. The unit vectors are also known as basis vectors. This is because any vector in real three-dimensional space can be written in terms of these unit/basis vectors. In some sense they are the basic components of any vector. Other basis vectors could be used, however, such as in spherical and cylindrical coordinates. When dealing with more abstract and/or complex vectors, it is often helpful to ask what one would do for an ordinary three-dimensional vector. For example, properties of unit vectors, dot products, etc. in three-dimensions are similar to the analogous constructions in higher dimensions.

The inner product, or dot product, for two real three-dimensional vectors,

${\vec {v}}=v_{x}{\hat {x}}+v_{y}{\hat {y}}+v_{z}{\hat {z}},\;\;{\vec {w}}=w_{x}{\hat {x}}+w_{y}{\hat {y}}+w_{z}{\hat {z}},$

can be computed as follows:

${\vec {v}}\cdot {\vec {w}}=v_{x}w_{x}+v_{y}w_{y}+v_{z}w_{z}.$

For the inner product of ${\vec {v}}\,\!$ with itself, we get the square of the magnitude of ${\vec {v}}\,\!$, denoted $|{\vec {v}}|^{2}\,\!$:

$|{\vec {v}}|^{2}={\vec {v}}\cdot {\vec {v}}=v_{x}v_{x}+v_{y}v_{y}+v_{z}v_{z}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}.$

If we want a unit vector in the direction of ${\vec {v}}\,\!$, we can simply divide it by its magnitude:

${\hat {v}}={\frac {\vec {v}}{|{\vec {v}}|}}.$

Now, of course, ${\hat {v}}\cdot {\hat {v}}=1\,\!$, which can easily be checked.

There are several ways to represent a vector. The ones we will use most often are column and row vector notations. So, for example, we could write the vector above as

${\vec {v}}=\left({\begin{array}{c}v_{x}\\v_{y}\\v_{z}\end{array}}\right).$

In this case, our unit vectors are represented by the following:

${\hat {x}}=\left({\begin{array}{c}1\\0\\0\end{array}}\right),\;\;{\hat {y}}=\left({\begin{array}{c}0\\1\\0\end{array}}\right),\;\;{\hat {z}}=\left({\begin{array}{c}0\\0\\1\end{array}}\right).\,\!$

We next turn to the subject of complex vectors and the relevant notation. We will see how to compute the inner product later, since some other definitions are required.

Complex Vectors

For complex vectors in quantum mechanics, Dirac notation is used most often. This notation uses a $\left\vert \cdot \right\rangle \,\!$, called a ket, for a vector. So our vector ${\vec {v}}\,\!$ would be

$\left\vert v\right\rangle =\left({\begin{array}{c}v_{x}\\v_{y}\\v_{z}\end{array}}\right).$

For qubits, i.e. two-state quantum systems, complex vectors will often be used:

	${\begin{aligned}\left\vert \psi \right\rangle &=\left({\begin{array}{c}\alpha \\\beta \end{array}}\right)\\&=\alpha \left\vert 0\right\rangle +\beta \left\vert 1\right\rangle ,\end{aligned}}$	(C.1)

where

$\left\vert 0\right\rangle =\left({\begin{array}{c}1\\0\end{array}}\right),\;\;{\mbox{and}}\;\;\left\vert 1\right\rangle =\left({\begin{array}{c}0\\1\end{array}}\right)$

are the basis vectors. The two numbers $\alpha \,\!$ and $\beta \,\!$ are complex numbers, so the vector is said to be a complex vector.

Inner Product

Now let us suppose we have another complex vector,

$\left\vert \phi \right\rangle =\left({\begin{array}{c}\gamma \\\delta \end{array}}\right).$

The inner product between two vectors is often written as $\left\langle \phi \vert \psi \right\rangle \;\!$, which is the same as

${\begin{aligned}(\left\vert \phi \right\rangle )^{\dagger }\left\vert \psi \right\rangle &=\left({\begin{array}{c}\gamma \\\delta \end{array}}\right)^{\dagger }\left({\begin{array}{c}\alpha \\\beta \end{array}}\right)\\&=\left({\begin{array}{cc}\gamma ^{*}&\delta ^{*}\end{array}}\right)\left({\begin{array}{c}\alpha \\\beta \end{array}}\right)\\&=\gamma ^{*}\alpha +\delta ^{*}\beta \end{aligned}}\;\!$

Outer Product

The outer product between these same two vectors is

${\begin{aligned}(\left\vert \phi \right\rangle )(\left\vert \psi \right\rangle )^{\dagger }&=\left\vert \phi \right\rangle \left\langle \psi \right\vert \\&=\left({\begin{array}{c}\gamma \\\delta \end{array}}\right)\left({\begin{array}{c}\alpha \\\beta \end{array}}\right)^{\dagger }\\&=\left({\begin{array}{c}\gamma \\\delta \end{array}}\right)\left({\begin{array}{cc}\alpha ^{*}&\beta ^{*}\end{array}}\right)\\&=\left({\begin{array}{cc}\gamma \alpha ^{*}&\gamma \beta ^{*}\\\delta \alpha ^{*}&\delta \beta ^{*}\end{array}}\right)\end{aligned}}\;\!$

Linear Algebra: Matrices

There are many aspects of linear algebra that are quite useful in quantum mechanics. We will briefly discuss several of these aspects here. First, some definitions and properties are provided that will be useful. Some familiarity with matrices will be assumed, although many basic definitions are also included.

Let us denote some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\times n\,\!} matrix by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!} . The set of all $m\times n\,\!$ matrices with real entries is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(n\times m,\mathbb{R})\,\!} . Such matrices are said to be real since they have all real entries. Similarly, the set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\times n\,\!} complex matrices is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(m\times n,\mathbb{C})\,\!} . For the set of set of square $n\times n\,\!$ complex matrices, we simply write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(n,\mathbb{C})\,\!} .

We will also refer to the set of matrix elements, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{ij}\,\!} , where the first index (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\,\!} in this case) labels the row and the second $(j)\,\!$ labels the column. Thus the element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{23}\,\!} is the element in the second row and third column. A comma is inserted if there is some ambiguity. For example, in a large matrix the element in the 2nd row and 12th column is written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{2,12}\,\!} to distinguish between the 21st row and 2nd column.

Complex Conjugate

The complex conjugate of a matrix is the matrix with each element replaced by its complex conjugate. In other words, to take the complex conjugate of a matrix, one takes the complex conjugate of each entry in the matrix. We denote the complex conjugate with a star, like this: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^*\,\!} . For example,

	${\begin{aligned}A^{*}&=&\left({\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}}\right)^{*}\\&=&\left({\begin{array}{ccc}a_{11}^{*}&a_{12}^{*}&a_{13}^{*}\\a_{21}^{*}&a_{22}^{*}&a_{23}^{*}\\a_{31}^{*}&a_{32}^{*}&a_{33}^{*}\end{array}}\right).\end{aligned}}\,\!$	(C.2)

(Notice that the notation for a matrix is a capital letter, whereas the entries are represented by lower case letters.)

Transpose

The transpose of a matrix is the same set of elements, but now the first row becomes the first column, the second row becomes the second column, and so on. Thus the rows and columns are interchanged. For example, for a square Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3\,\!} matrix, the transpose is given by

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} A^T &=& \left(\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right)^T \\ &=& \left(\begin{array}{ccc} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32} \\ a_{13} & a_{23} & a_{33} \end{array}\right). \end{align} \,\!}	(C.3)

Hermitian Conjugate

The complex conjugate and transpose of a matrix is called the Hermitian conjugate, or simply the dagger of a matrix. It is called the dagger because the symbol used to denote it, (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dagger\,\!} ):

	$(A^{T})^{*}=(A^{*})^{T}\equiv A^{\dagger }.\,\!$	(C.4)

For our Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3\,\!} example,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^\dagger = \left(\begin{array}{ccc} a_{11}^* & a_{21}^* & a_{31}^* \\ a_{12}^* & a_{22}^* & a_{32}^* \\ a_{13}^* & a_{23}^* & a_{33}^* \end{array}\right). \,\!}

If a matrix is its own Hermitian conjugate, i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^\dagger = A\,\!} , then we call it a Hermitian matrix. (Clearly this is only possible for square matrices.) Hermitian matrices are very important in quantum mechanics since their eigenvalues are real. (See Sec.(Eigenvalues and Eigenvectors).)

Index Notation

Very often we write the product of two matrices $A\,\!$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\,\!} simply as $AB\,\!$ and let $C=AB\,\!$. However, it is also quite useful to write this in component form. In this case, if these are $n\times n\,\!$ matrices

$c_{ik}=\sum _{j=1}^{n}a_{ij}b_{jk}.\,\!$

This says that the element in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{\mbox{th}}\,\!} row and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{\mbox{th}}\,\!} column of the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\,\!} is the sum $\sum _{1}^{n}a_{ij}b_{jk}\,\!$. The transpose of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\,\!} has elements

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}. \,\!}

Now if we were to transpose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!} and $B\,\!$ as well, this would read

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_1^n b^T_{ij} a^T_{jk}. \,\!}

This gives us a way of seeing the general rule that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^T = B^TA^T. \,\!}

It follows that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^\dagger = B^\dagger A^\dagger. \,\!}

The Trace

The trace of a matrix is the sum of the diagonal elements and is denoted ${\mbox{Tr}}\,\!$. So for example, the trace of an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n\,\!} matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!} is

${\mbox{Tr}}(A)=\sum _{i=1}^{n}a_{ii}\,\!$

Some useful properties of the trace are the following:

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!}
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!}

Using the first of these results,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A). \,\!}

This relation is used so often that we state it here explicitly.

The Determinant

For a square matrix, the determinant is quite a useful thing. For example, an $n\times n\,\!$ matrix is invertible if and only if its determinant is not zero. So let us define the determinant and give some properties and examples.

The determinant of a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\times 2\,\!} matrix

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = \left(\begin{array}{cc} a & b \\ c & d \end{array}\right), \,\!}	(C.5)

is given by

	$\det(N)=ad-bc.\,\!$	(C.6)

Higher-order determinants can be written in terms of smaller ones in the standard way.

The determinant of a matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!} can be also be written in terms of its components as

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...} a_{1i}a_{2j}a_{3k}a_{4l} ..., \,\!}	(C.7)

where the symbol

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ijkl...} = \begin{cases} +1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\ -1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\ \;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}). \end{cases} \,\!}	(C.8)

Let us consider the example of the $3\times 3\,\!$ matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!} given above. The determinant can be calculated by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(A) = \sum_{i,j,k} \epsilon_{ijk} a_{1i}a_{2j}a_{3k}, \,\!}

where, explicitly,

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ijk} = \begin{cases} +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\ -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\ \;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}). \end{cases} \,\!}	(C.9)

so that

	${\begin{aligned}\det(A)&=&\epsilon _{123}a_{11}a_{22}a_{33}+\epsilon _{132}a_{11}a_{23}a_{32}+\epsilon _{231}a_{12}a_{23}a_{31}\\&&+\epsilon _{213}a_{12}a_{21}a_{33}+\epsilon _{312}a_{13}a_{21}a_{32}+\epsilon _{213}a_{13}a_{21}a_{32}.\end{aligned}}\,\!$	(C.10)

Now given the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ijk}\,\!} in Eq. C.9, this is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} - a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{21}a_{32}. \,\!}

The determinant has several properties which are useful to know. A few are listed here.

1. The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(A) = \det(A^T).\,\!}
2. The determinant of a product is the product of determinants: $\det(AB)=\det(A)\det(B).\,\!$

From this last property, another specific property can be derived. Suppose we take the determinant of the product of a matrix and its inverse we find

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1, \,\!}

since the determinant of the identity is one. This implies that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(U^{-1}) = \frac{1}{\det(U)}. \,\!}

The Inverse of a Matrix

The inverse of a square matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!} is another matrix, denoted $A^{-1}\,\!$ such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AA^{-1} = A^{-1}A = \mathbb{I}, \,\!}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{I}\,\!} is the identity matrix consisting of zeroes everywhere except the diagonal which has ones. For example the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3\,\!} identity matrix is

$\mathbb {I} _{3}=\left({\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}}\right).\,\!$

It is important to note that a matrix is invertible if and only if its determinant is nonzero. Thus one only needsd to calculate the determinant to see if a matrix has an inverse or not.

Hermitian Matrices

Hermitian matrices are important for a variety of reasons, but primarily they are important because their eigenvalues are real. Therefore Hermitian matrices are used to represent density operators and density matrices, and also Hamiltonians. The density operator is a positive semidefinite Hermitian matrix (it has no negative eigenvalues) which has its trace equal to one. In either case it is often desirable to represent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\times N\,\!} Hermitian matrices using a real linear combination of a complete set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\times N\,\!} Hermitian matrices. A set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\times N\,\!} Hermitian matrices is complete if any Hermitian matrix can be represented in terms of the set. Let $\{\lambda _{i}\}\,\!$ be a complete set. Then any Hermitian matrix can be represented by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_i a_i \lambda_i\,\!} . The set can always be taken to be a set of traceless Hermitian matrices and the identity matrix. This is convenient for the density matrix since its trace is one because the identity part of an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\times N\,\!} Hermitian matrix is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1/N)\mathbb{I}\,\!} if we take all others in the set to be traceless. For the Hamiltonian, the set consists of a traceless part and an identity part and the identity part just gives an overall phase which can often be neglected.

One example of such a set which is extremely useful is the set of Pauli matrices. These are discussed in detail in Chapter 2 and in particular in Section 2.4.

Unitary Matrices

A unitary matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\,\!} is one whose inverse is also its Hermitian conjugate, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U^\dagger = U^{-1}\,\!} so that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U^\dagger U = UU^\dagger = \mathbb{I}. \,\!}

If the unitary matrix also has determinant one, it is said to be a special unitary matrix. The set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n\,\!} unitary matrices is denoted UFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n)\,\!} and the set of special unitary matrices is denoted SUFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n)\,\!} .

Unitary matrices are particularly important in quantum mechanics because they describe the evolution, or change, of quantum states. They are able to do this because unitary matrices have the property that rows and columns, viewed as vectors, are orthonormal. (To see this, an example is provided below.) This means that when they act on a basis vector of the form (one 1, in say the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} th spot, and zeroes everywhere else)

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert j\right\rangle = \left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 \end{array}\right), \,\!}	(C.11)

the result is a normalized complex vector. Acting on a set of orthonormal vectors of the form given in Eq.(C.11), will produce another orthonormal set.

Let us consider the example of a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\times 2\,\!} unitary matrix,

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = \left(\begin{array}{cc} a & b \\ c & d \end{array}\right). \,\!}	(C.12)

The inverse of this matrix is the Hermitian conjugate, so the inverse is given by

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U ^{-1} = U^\dagger = \left(\begin{array}{cc} a^* & c^* \\ b^* & d^* \end{array}\right), \,\!}	(C.13)

provided that the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\,\!} satisfies the constraints

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} |a|^2 + |b|^2 = 1, \; & \; ac^*+bd^* =0 \\ ca^*+db^*=0, \; & \; |c|^2 + |d|^2 =1, \end{align}\,\!}	(C.14)

and

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} |a|^2 + |c|^2 = 1, \; & \; ba^*+dc^* =0 \\ b^*a+d^*c=0, \; & \; |b|^2 + |d|^2 =1. \end{align}\,\!}	(C.15)

Looking at each row as a vector, the constraints in Eq.(C.14) are the orthonormality conditions for the vectors forming the rows. Similarly, the constraints in Eq.(C.15) are the orthonormality conditions for the vectors forming the columns when viewed as vectors.

More Dirac Notation

Now that we have a definition of Hermitian conjugate, we consider the case for a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\times n\,\!} matrix, i.e. a vector. In Dirac notation, we had

$\left\vert \psi \right\rangle =\left({\begin{array}{c}\alpha \\\beta \end{array}}\right),\,\!$

So the Hermitian conjugate comes up so often that we use the following notation for vectors,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\ \beta \end{array}\right)^\dagger = \left( \alpha^*, \; \beta^* \right). \,\!}

This is a row vector and in Dirac notation, the symbol Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle\cdot \right\vert\!} , is called a bra. Let us consider a second complex vector

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta \end{array}\right). \,\!}

The inner product between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\psi\right\rangle\,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\phi\right\rangle\,\!} is computed as follows:

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left\langle\phi\mid\psi\right\rangle & \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle \\ &= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta \end{array}\right) \\ &= \gamma^*\alpha + \delta^*\beta. \end{align} }	(C.16)

If these two vectors are orthogonal, then their inner product is zero, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle\phi\mid\psi\right\rangle =0\,\!} . (The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle\phi\mid\psi\right\rangle \,\!} is called a bracket which is the product of the bra and the ket.) The inner product of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\psi\right\rangle\,\!} with itself is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2. \,\!}

If this vector is normalized then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle\psi\mid\psi\right\rangle = 1\,\!} .

More generally, we will consider vectors in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\,\!} dimensions. In this case we write the vector in terms of a set of basis vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\left\vert i\right\rangle\}\,\!} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 0,1,2,...N-1\,\!} . This is an ordered set of vectors which are just labeled by integers. If the set is orthogonal, then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j, \,\!}

and if they are normalized, then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i. \,\!}

If both of these are true, i.e., the entire set is orthonormal, we can write,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle i\mid j\right\rangle = \delta_{ij}, \,\!}

where the symbol $\delta _{ij}\,\!$ is called the Kronecker delta and is defined by

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{ij} = \begin{cases} 1, & \mbox{if } i=j, \\ 0, & \mbox{if } i\neq j. \end{cases} }	(C.17)

Now consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (N+1)\,\!} -dimensional vectors by letting two such vectors be expressed in the same basis as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle, \,\!}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle. \,\!}

Then the inner product is

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left\langle\Psi\mid\Phi\right\rangle &= \left(\sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right) \\ &= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle \\ &= \sum_{ij} \alpha_i^*\beta_j\delta_{ij} \\ &= \sum_i\alpha^*_i\beta_i, \end{align} }	(C.18)

where we have used the fact that the delta function is zero unless Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=j\,\!} to get the last equality. For the inner product of a vector with itself, we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2. \,\!}

This immediately gives us a very important property of the inner product. It tells us that in general,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0 \Leftrightarrow \left\vert\Phi\right\rangle = 0. \,\!}

(Just in case you don't know, the symbol Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftrightarrow\,\!} means "if and only if" sometimes written as "iff.")

We could also expand a vector in a different basis. Let us suppose that the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\left\vert e_k \right\rangle\}\,\!} is an orthonormal basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\left\langle e_k \mid e_l\right\rangle = \delta_{kl})\,\!} which is different from the one considered earlier. We could expand our vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\Psi\right\rangle\,\!} in terms of our new basis by expanding our new basis in terms of our old basis. Let us first expand the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert e_k\right\rangle\,\!} in terms of the $\left\vert j\right\rangle \,\!$:

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle, }	(C.19)

so that

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left\vert \Psi\right\rangle &= \sum_j \alpha_j\left\vert j\right\rangle \\ &= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle \\ &= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, \end{align} }	(C.20)

where

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. }	(C.21)

Notice that the insertion of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!} didn't do anything to our original vector. It is the same vector, just in a different basis. Therefore, this is effectively the identity operator

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert. \,\!}

This is an important and quite useful relation. Now, to interpret Eq.(C.19), we can draw a close analogy with three-dimensional real vectors. The inner product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert e_k \right\rangle \left\langle j \right\vert\,\!} can be interpreted as the projection of one vector onto another. This provides the part of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert j \right\rangle\,\!} along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert e_k \right\rangle\,\!} .

Transformations

Suppose we have two different orthogonal bases, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{e_k\}\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{j\}\,\!} . The numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle e_k\mid j\right\rangle\,\!} for all the different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j\,\!} are often referred to as matrix elements since the set forms a matrix with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\,\!} labelling the rows, and $j\,\!$ labelling the columns. Therefore, we can write the transformation from one basis to another with a matrix transformation. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\,\!} be the matrix with elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{kj} = \left\langle e_k\mid j\right\rangle\,\!} . Then the transformation from one basis to another, written in terms of the coefficients of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\Psi\right\rangle\,\!} , is

	$A^{\prime }=MA,$	(C.22)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\ \alpha_n^\prime \end{array}\right), \;\; \mbox{ and } \;\; A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_n\end{array}\right). \,\!}

This sort of transformation is a change of basis. However, often when one vector is transformed to another, the transformation can be viewed as a transformation of the components of the vector and is also represented by a matrix. Thus transformations can either be represented by the matrix equation, like Eq.(C.22), or the components

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. }	(C.23)

In the case that we consider a matrix transformation of basis elements, we call it a passive transformation. (The transformation does nothing to the object, but only changes the basis in which the object is described.) An active transformation is one where the object itself is transformed. Often these two transformations, active and passive, are very simply related. However, the distinction can be very important.

For a general transformation matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T\,\!} , acting on a vector, the matrix elements in a particular basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert i\right\rangle\,\!} are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, \,\!}

just as elements of a vector can be found using

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle i\mid \Psi \right\rangle = \alpha_i. \,\!}

Transformations of a Qubit

It is worth belaboring the point somewhat and presenting several ways in which to parametrize the set of transformations of a qubit. A qubit state is represented by a complex two-dimensional vector that has been normalized to one:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert \psi\right\rangle = \alpha_0 \left\vert 0 \right\rangle + \alpha_1 \left\vert 1\right\rangle = \left(\begin{array}{c} \alpha_0 \\ \alpha_1 \end{array}\right), \;\;\;\; |\alpha_0|^2 + |\alpha_1|^2 = 1. \,\!}

The most general matrix transformation that will take this to any other state of the same form (complex, 2-d vector with unit norm) is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\times 2\,\!} unitary matrix. In Chapter 2 several specific examples of qubit transformations were given and in Chapter 3, Section 3.4 it was stated that an element of SU(2) can be written as (see Section 3.2.1, Exponentiation of a Matrix)

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} U(\theta) &= \exp(-i\vec{n}\cdot\vec{\sigma} \theta/2) \\ &= (\mathbb{I}\cos(\theta/2) -i\vec{n}\cdot\vec{\sigma} \sin(\theta/2)) \end{align} }	(C.24)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{n}\,\!} is a unit vector, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\vec{n}|=1\,\!} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{n}\cdot\vec{\sigma} = n_1\sigma_1+n_2\sigma_2+n_3\sigma_3\,\!} . Explicitly, this is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \exp(-i\vec{n}\cdot\vec{\sigma} \theta/2) &= \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)\cos(\theta/2) \\ & \;\;\; + (-i)\left[ n_1\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) + n_2\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right) + n_3\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)\right]\sin(\theta/2) \\ &= \left(\begin{array}{cc} \cos(\theta/2) -in_3\sin(\theta/2) & (-in_1-n_2)\sin(\theta/2) \\ (-in_1+n_2)\sin(\theta/2) & \cos(\theta/2) +in_3\sin(\theta/2) \end{array}\right). \end{align} }

Notice that this is a special unitary matrix. (See Section Unitary Matrices.) To see that this is the most general SU(2) matrix, one needs to verify that any complex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\times 2\,\!} unitary matrix can be written in this form. (One way to do that is to start with a generic matrix and impose the restrictions. Here one may simply convince oneself that this is general by observation by acting on basis vectors.) This is the most general qubit transformation and can be interpreted as a rotation about the axis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{n}\,\!} by an angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta\,\!} .

Another parametrization of this set of matrices is the following, called the Euler angle parametrization,

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{EA} = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2). \,\!}	(C.25)

In this case the matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_z\,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_y\,\!} are not unique. Any two of the three Pauli matrices (or one of each) may be chosen. This is quite simple, useful, and generalizable to SU(N) for N arbitrary. In the simple case of a qubit, one may convince oneself by acting on basis vectors as before. However, with a little thought, one may see that rotating to a position on the sphere by the first angle, followed by rotations using the other two, will provide for a general orientation of an object.

Similarity Transformation

A similarity transformation of an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n\,\!} matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!} by an invertible matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\,\!} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S A S^{-1}\,\!} . There are (at least) two important things to note about similarity transformations,

1. Similarity transformations leave the trace of a matrix unchanged. This is shown explicitly in Section 3.5
2. Similarity transformations leave the determinant of a matrix unchanged. (We say the determinant is invariant under similarity transformations.) This is because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!}
3. Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^\prime = SAS^{-1}\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B^\prime = SBS^{-1}\,\!} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^\prime = SCS^{-1}\,\!} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB=C\,\!} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^\prime B^\prime = C^\prime\,\!} since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} = SCS^{-1}=C^\prime\,\!} . The two matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^\prime\,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!} are said to be similar.

Eigenvalues and Eigenvectors

A matrix can always be diagonalized. By this, it is meant that for every complex matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\,\!} there is a diagonal matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D\,\!} such that

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = UDV, \,\!}	(C.26)

where $U\,\!$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\,\!} are unitary matrices. This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D\,\!} are called the singular values of the matrix $M\,\!$. However, the singular values are not always easy to find.

For the special case that the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\,\!} is Hermitian Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M^\dagger = M)\,\!} , the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\,\!} can be written as

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = U D U^\dagger, }	(C.27)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\,\!} is unitary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (U^{-1}=U^\dagger)\,\!} . In this case the elements of the matrix $D\,\!$ are called eigenvalues. Very often eigenvalues are introduced as solutions to the equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle \,\!}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert v\right\rangle\,\!} a vector called an eigenvector.

To find the eigenvalues and eigenvectors of a matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\,\!} , we follow a standard procedure which is to calculate the following

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(\lambda\mathbb{I} - M) = 0, }	(C.28)

and solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda\,\!} . The different solutions for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda\,\!} is the set of eigenvalues and this set is called the spectrum. Let the different eigenvalues be denoted by $\lambda _{i}\,\!$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1,2,...,n\,\!} fo an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n\,\!} vector. If two eigenvalues are equal, we say the spectrum is degenerate. To find the eigenvectors, which correspond to different eigenvalues, the equation

$M\left\vert v\right\rangle =\lambda _{i}\left\vert v\right\rangle \,\!$

must be solved for each value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\,\!} . Notice that this equations holds even if we multiply both sides by some complex number. This implies that an eigenvector can always be scaled. Usually they are normalized to obtain an orthonormal set. As we will see by example, degenerate eigenvalues require some care.

Example 1

Consider a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\times 2\,\!} Hermitian matrix

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \left(\begin{array}{cc} 1+a & b-ic \\ b+ic & 1-a \end{array}\right). }	(C.29)

To find the eigenvalues of this, we follow a standard procedure which is to calculate the following

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(\sigma-\lambda\mathbb{I}) = 0, }	(C.30)

and solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda\,\!} . The eigenvalues of this matrix are given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det\left(\begin{array}{cc} 1+a-\lambda & b-ic \\ b+ic & 1-a-\lambda \end{array}\right) =0, \,\!}

which implies the eigenvalues are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}. \,\!}

and the eigenvectors are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1=\left(\begin{array}{c} i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ 2b \end{array}\right), v_2= \left(\begin{array}{c} i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ 2b \end{array}\right). \,\!}

These expressions are useful for calculating properties of qubit states as will be seen in the text.

Example 2

Now consider a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3\,\!} matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N= \left(\begin{array}{ccc} 1 & -i & 0 \\ i & 1 & 0 \\ 0 & 0 & 1 \end{array}\right). \,\!}

First we calculate

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det\left(\begin{array}{ccc} 1-\lambda & -i & 0 \\ i & 1-\lambda & 0 \\ 0 & 0 & 1-\lambda \end{array}\right) = (1-\lambda)[(1-\lambda)^2-1]. \,\!}

This implies that the eigenvalues C.6 are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = 1,0, \mbox{ or } 2. \,\!}

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_1=1\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_0 = 0\,\!} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_2 = 2\,\!} . To find eigenvectors, we calculate

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} Nv &= \lambda v, \\ \left(\begin{array}{ccc} 1 & -i & 0 \\ i & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array}\right) &= \lambda\left(\begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array}\right). \end{align} }	(C.31)

for each $\lambda \,\!$. For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = 1\,\!} we get the following equations:

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} v_1 -iv_2 &= v_1, \\ iv_1+v_2 &= v_2, \\ v_3 &= v_3, \end{align} }	(C.32)

so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_2 =0\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1 =0\,\!} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_3\,\!} is any non-zero number, but we choose it to normalize the vector. For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda =0\,\!} ,

	${\begin{aligned}v_{1}&=iv_{2},\\v_{3}&=0,\end{aligned}}$	(C.33)

and for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = 2\,\!} ,

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} v_1 -iv_2 &= 2v_1, \\ iv_1+v_2 &= 2v_2, \\ v_3 &= 2v_3, \end{align} }	(C.34)

so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1 = -iv_2\,\!} . Therefore, our three eigenvectors are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right). \,\!}

The matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V= (v_0,v_1,v_2) = \left(\begin{array}{ccc} i/\sqrt{2} & 0 & -i/\sqrt{2} \\ 1/\sqrt{2} & 0 & 1/\sqrt{2} \\ 0 & 1 & 0 \end{array}\right) \,\!}

is the matrix that diagonalizes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\,\!} in the following way,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = VDV^\dagger, \,\!}

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}\right) \,\!}

or, we may write this as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^\dagger N V = D. \,\!}

This is sometimes called the eigenvalue decompostion of the matrix and is also written as,

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = \sum_i \lambda_i v_iv^\dagger_i. \,\!}	(C.35)

Example 3

Next, consider a complex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3} Hermitian matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = \left(\begin{array}{ccc} \frac{5}{2} & 0 & \frac{i}{2} \\ 0 & 2 & 0 \\ -\frac{i}{2} & 0 & \frac{5}{2} \end{array}\right). \,\!}

First we calculate

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det\left(\begin{array}{ccc} \frac{5}{2}-\lambda & 0 & \frac{i}{2} \\ 0 & 2-\lambda & 0 \\ -\frac{i}{2} & 0 & \frac{5}{2}-\lambda \end{array}\right) = (2-\lambda)\left[\left(\frac{5}{2}-\lambda\right)^2-\frac{1}{4}\right]. \,\!}

This implies that the eigenvalues C.6 are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = 2,2, \mbox{ or } 3. \,\!}

Note that there are two which are the same. These are called degenerate. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_1=2\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_2 = 2\,\!} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_3 = 3\,\!} . To find eigenvectors, we calculate

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} Mv &= \lambda v, \\ \left(\begin{array}{ccc} \frac{5}{2} & 0 & \frac{i}{2} \\ 0 & 2 & 0 \\ -\frac{i}{2} & 0 & \frac{5}{2} \end{array}\right)\left(\begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array}\right) &= \lambda\left(\begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array}\right). \end{align} }	(C.36)

for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda\,\!} . For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = 3\,\!} we get the following equations:

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{5}{2}v_1 + \frac{i}{2}v_3 &= 3v_1, \\ 2v_2 &= 3v_2, \\ -\frac{i}{2}v_1 + \frac{5}{2}v_3 &= 3v_3, \end{align} }	(C.37)

so

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} iv_3 &= v_1, \\ v_2 &= 0, \end{align} }	(C.38)

Now for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = 2\,\!} ,

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{5}{2}v_1 +\frac{i}{2}v_3 &= 2 v_1, \\ 2v_2 &= 2v_2, \\ -\frac{i}{2}v_1 + \frac{5}{2}v_3 &= 2v_3, \end{align} }	(C.39)

so

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} v_3 &= iv_1, \\ v_2 &= \mbox{anything}, \end{align} }	(C.40)

We would like to have a set of orthonormal vectors and we can always choose the set to be orthonormal. So we choose the three eigenvectors to be

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_2 = \left(\begin{array}{c} 1 \\ a \\ i \end{array}\right), \;\; v_2^\prime = \left(\begin{array}{c} 1 \\ a^\prime \\ i \end{array}\right), \;\; v_3 = \left(\begin{array}{c} i \\ 0 \\ 1 \end{array}\right), \,\!}

and we set the inner product of the two vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_2\,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_2^\prime \,\!} equal to zero so as to have then be orthogonal,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + a a^\prime +1 = 2 + a a^\prime = 0. \,\!}

Now we can choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = \sqrt{2}\,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^\prime = -\sqrt{2}\,\!} so that the normalized eigenvectors are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_2 = \frac{1}{2}\left(\begin{array}{c} 1 \\ \sqrt{2} \\ i \end{array}\right), \;\; v_2^\prime = \frac{1}{2}\left(\begin{array}{c} 1 \\ -\sqrt{2} \\ i \end{array}\right), \;\; v_3 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 0 \\ 1 \end{array}\right). \,\!}

Tensor Products

The tensor product (also called the Kronecker product) is used extensively in quantum mechanics and throughout the course. It is commonly denoted with a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \otimes\,\!} symbol, but this symbol is also often left out. In fact the following are commonly found in the literature as notation for the tensor product of two vectors $\left\vert \Psi \right\rangle \,\!$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\Phi\right\rangle\,\!}

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle \\ &= \left\vert\Psi\Phi\right\rangle. \end{align} }	(C.41)

Each of these has its advantages and we will use all of them in different circumstances.

The tensor product is also often used for operators. So several examples will be given, one which explicitly calculates the tensor product for two vectors and one which calculates it for two matrices which could represent operators. However, these are not different in the sense that a vector is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\times n\,\!} or an $n\times 1\,\!$ matrix. It is also noteworthy that the two objects in the tensor product need not be of the same type. In general a tensor product of an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times m\,\!} object (array) with a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\times q\,\!} object will produce an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle np\times mq\,\!} object.

In general, the tensor product of two objects is computed as follows. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!} be an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times m\,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\,\!} be a $p\times q\,\!$ array

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \left(\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1m} \\ a_{21} & a_{22} & \cdots & a_{2m} \\ \vdots & & \ddots & \\ a_{n1} & a_{n2} & \cdots & a_{nm} \end{array}\right), }	(C.42)

and similarly for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\,\!} . Then

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\otimes B = \left(\begin{array}{cccc} a_{11}B & a_{12}B & \cdots & a_{1m}B \\ a_{21}B & a_{22}B & \cdots & a_{2m}B \\ \vdots & & \ddots & \\ a_{n1}B & a_{n2}B & \cdots & a_{nm}B \end{array}\right). }	(C.43)

Let us now consider two examples. First let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\phi\right\rangle\,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\psi\right\rangle\,\!} be as before,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta \end{array}\right), \;\; \mbox{and} \;\; \left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta \end{array}\right). \,\!}

Then

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &= \left(\begin{array}{c} \alpha \\ \beta \end{array}\right) \otimes \left(\begin{array}{c} \gamma \\ \delta \end{array}\right) \\ &= \left(\begin{array}{c} \alpha\gamma\\ \alpha\delta \\ \beta\gamma \\ \beta\delta \end{array}\right). \end{align} }	(C.44)

Also

	${\begin{aligned}\left\vert \psi \right\rangle \otimes \left\langle \phi \right\vert &=\left\vert \psi \right\rangle \left\langle \phi \right\vert \\&=\left({\begin{array}{c}\alpha \\\beta \end{array}}\right)\otimes \left({\begin{array}{cc}\gamma ^{*}&\delta ^{*}\end{array}}\right)\\&=\left({\begin{array}{cc}\alpha \gamma ^{*}&\alpha \delta ^{*}\\\beta \gamma ^{*}&\beta \delta ^{*}\end{array}}\right).\end{aligned}}$	(C.45)

Now consider two matrices

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \left(\begin{array}{cc} a & b \\ c & d \end{array}\right), \;\; \mbox{and} \;\; B = \left(\begin{array}{cc} e & f \\ g & h \end{array}\right). \,\!}

Then

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} A\otimes B &= \left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \otimes \left(\begin{array}{cc} e & f \\ g & h \end{array}\right) \\ &= \left(\begin{array}{cccc} ae & af & be & bf \\ ag & ah & bg & bh \\ ce & cf & de & df \\ cg & ch & dg & dh \end{array}\right). \end{align} }	(C.46)

Properties of Tensor Products

Some properties of tensor products which are useful are the following (with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\,\!} , $C\,\!$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D\,\!} any type):

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A\otimes B)(C\otimes D) = AC \otimes BD\,\!}
2. $(A\otimes B)^{T}=A^{T}\otimes B^{T}\,\!$
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A\otimes B)^* = A^*\otimes B^*\,\!}
4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!}
5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A+B) \otimes C = A\otimes C+B\otimes C\,\!}
6. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\otimes(B+C) = A\otimes B + A\otimes C\,\!}
7. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!}

(See Horn and Johnson, Topics in Matrix Analysis, Chapter 4.)