View source for Matrices and Vectors: A new beginning

=Vectors=

Here we introduce vectors and the notation that we use for vectors.  We then give some facts about real vectors before discussing the complex vectors used in quantum mechanics. 

===Vectors Defining and Representing===

You may have heard of the definition of a vector as a quantity with both magnitude and direction.  While this is true and often used in science classes, our purpose is different.  So we will simply define a vector as a set of numbers that is written in an row or a column.  When the vector is written as a row of numbers, it is called a row vector and when it is written as a set of numbers in a column, it is called a column vector.  As we will see, these two can have the same set of numbers, but each can be used in a slightly different way.  

====Examples====


This is an example of a row vector
<math>\vec{v} = (2,4,3). </math>

This is an example of a column vector
<math>\vec{w} = \left(\begin{array}{c} 1 \\ 5 \\ 4 \end{array}\right).</math>

===Real Vectors===


If you are familiar with 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.  However, this is not necessary and we will see how to perform all of the operations that we need just using our arrays of numbers.  In three dimensional space,
a vector is often written as
<center><math>
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},
</math></center>
where the hat (<math>\hat{\cdot}\,\!</math>) denotes a unit vector and the components
<math>v_i\,\!</math>, <math>i = x,y,z\,\!</math> are just numbers.  The unit vectors are also
known as ''basis'' vectors. <!-- \index{basis vectors!real} -->  The unit vectors have ''magnitude'' equal to one.  (The magnitude is the size, or length of a vector.)  So that if the vector <math>\hat{v}\,\!</math> is a unit vector, then <math>v_x^2 + v_y^2 + v_z^2 =1\,\!</math>

===Vector Operations===

====Adding Vectors====

When adding vectors, it is important to note that you can only add vectors if they are the same type.  (No adding "apples and oranges" so to speak.)  So you can add two vectors that are both column vectors and have three entries.  You can't add a column vector to a row vector and you can't add a vector with two components to a vector with three components.


====Example====

Adding two vectors <math>\vec{v}\,\!</math> and <math>\vec{w}\,\!</math> with 
<center><math>
\vec{v} = \left(\begin{array}{c} 1 \\ 5
  \end{array}\right), \;\;  
\vec{w} = \left(\begin{array}{c} 4 \\ 2
  \end{array}\right),
\,\!</math></center>
we get 
<center><math>
\vec{v} + \vec{w} = \left(\begin{array}{c} 1 \\ 5
  \end{array}\right) + \left(\begin{array}{c} 4 \\ 2
  \end{array}\right) = \left(\begin{array}{c} 5 \\ 7
  \end{array}\right).
\,\!</math></center>

====Products of Vectors====

The ''inner product'',<!-- \index{inner product}--> or ''dot product'',<!--\index{dot product}--> for two real three-dimensional vectors,
<center><math>
\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},
</math></center>
can be computed as follows:
<center><math>
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.
</math></center>
For the inner product of <math>\vec{v}\,\!</math> with itself, we get the square of
the magnitude of <math>\vec{v}\,\!</math>, denoted <math>|\vec{v}|^2\,\!</math>:
<center><math>
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +
v_zv_z=v_x^2+v_y^2+v_z^2. 
</math></center>
If we want a unit vector in the direction of <math>\vec{v}\,\!</math>, we can simply divide it
by its magnitude:
<center><math>
\hat{v} = \frac{\vec{v}}{|\vec{v}|}.  
</math></center>
Now, of course, <math>\hat{v}\cdot\hat{v}= 1\,\!</math>, 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
<center><math>
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z
  \end{array}\right).  
</math></center>
In this case, our unit vectors are represented by the following: 
<center><math>
\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).
\,\!</math></center>

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 <math>\left\vert \cdot \right\rangle\,\!</math>, 
called a ''ket'', for a vector.  So our vector <math>\vec{v}\,\!</math> would be
<center><math>
\left\vert v \right\rangle  = \left(\begin{array}{c} v_x \\ v_y \\ v_z
  \end{array}\right).  
</math></center>

For qubits, i.e. two-state quantum systems, complex vectors will often be used:
{{Equation|<math>\begin{align} \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{align}</math>|C.1}} 
where
<center><math>
\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)
</math></center>
are the basis vectors.  The two numbers <math>\alpha\,\!</math> and
<math>\beta\,\!</math> are complex numbers, so the vector is said to
be a complex vector.

=Matrices=


===Basic Definition and Representations===

A matrix is an array of numbers of the following form with columns, col. 1, col. 2, etc., and rows, row 1, row 2, etc.  The entries for the matrix are labeled by the row and column.  So the entry of a matrix <math>A</math> will be <math>a_{ij}</math> where <math>i</math> is the row and <math>j</math> is the column where the number <math>a_{ij}</math> is found.  This is how it looks:

<center><math>
A = \begin{array}{c} {\scriptstyle{row \; 1}}  \\ {\scriptstyle{row \; 2}}  \\  \vdots  \\  {\scriptstyle{row \; m}}\end{array} \overset{{\scriptstyle{col. \; 1 }} \;\;\; {\scriptstyle{col.\; 2 }} \;\;\;\; {\displaystyle{\cdots}} \;\;\;\;\; {\scriptstyle{col.\; n }}}{\left(\begin{array}{cccc} 
a_{11} & a_{12} & \cdots & a_{1n} \\ 
a_{21} & a_{22} & \cdots & a_{2n} \\ 
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}  
\end{array}\right)}.
\,\!</math></center>

Notice that we represent the whole matrix with a capital letter <math>A</math>.  We could also represent it using all of the entries, this array of numbers seen in the equation above.  Another way to represent it is to write it as <math>(a_{ij})</math>.  By this we mean that it is the array of numbers in the parentheses.  

===Matrix Addition===

Matrix addition is performed by adding each element of one matrix with the corresponding element in another matrix.  Let our two matrices be <math>A</math> as above, and <math>B</math>.  To represent these in an array, 
<center><math>
A = \left(\begin{array}{cccc} 
a_{11} & a_{12} & \cdots & a_{1n} \\ 
a_{21} & a_{22} & \cdots & a_{2n} \\ 
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}  
\end{array}\right), \;\;\;\;
B = \left(\begin{array}{cccc} 
b_{11} & b_{12} & \cdots & b_{1n} \\ 
b_{21} & b_{22} & \cdots & b_{2n} \\ 
\vdots & \vdots & \ddots & \vdots \\
b_{m1} & b_{m2} & \cdots & b_{mn}  
\end{array}\right).
\,\!</math></center>


The the sum, which we could call <math>C= A+B</math> is given by 
<center><math>
A + B = \left(\begin{array}{cccc} 
a_{11} & a_{12} & \cdots & a_{1n} \\ 
a_{21} & a_{22} & \cdots & a_{2n} \\ 
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}  
\end{array}\right) 
+ 
\left(\begin{array}{cccc} 
b_{11} & b_{12} & \cdots & b_{1n} \\ 
b_{21} & b_{22} & \cdots & b_{2n} \\ 
\vdots & \vdots & \ddots & \vdots \\
b_{m1} & b_{m2} & \cdots & b_{mn}  
\end{array}\right) 
=
\left(\begin{array}{cccc} 
a_{11} + b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+b_{1n} \\ 
a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2n}+b_{2n} \\ 
\vdots & \vdots & \ddots & \vdots \\
a_{m1}+b_{m1} & a_{m2}+b_{m2} & \cdots & a_{mn}+b_{mn}  
\end{array}\right). 
\,\!</math></center>
In other words, the sum gives <math>c_{11} = a_{11} + b_{11}\,\!</math>, etc.  We add them component by component like we do vectors.  


===Notation===

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 <math>m\times n\,\!</math> matrix by <math>A\,\!</math>.  The set of all <math>m\times
n\,\!</math> matrices with real entries is <math>M(m\times n,\mathbb{R})\,\!</math>.  Such matrices
are said to be real since they have all real entries.  Similarly, the
set of <math>m\times n\,\!</math> complex matrices is <math>M(m\times n,\mathbb{C})\,\!</math>.  For the
set of square <math>n\times n\,\!</math> complex matrices, we simply write
<math>M(n,\mathbb{C})\,\!</math>.  


We will also refer to the set of matrix elements, <math>a_{ij}\,\!</math>, where the
first index (<math>i\,\!</math> in this case) labels the row and the second <math>(j)\,\!</math>
labels the column.  Thus the element <math>a_{23}\,\!</math> 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 <math>a_{2,12}\,\!</math> to distinguish between the
21st row and 2nd column.  


===The Identity Matrix===

An identity matrix has the property that when it is multiplied by any matrix, that matrix is unchanged.  That is, for any matrix <math>A\,\!</math>, 
<center><math>
\mathbb{I}A = A\mathbb{I} = A.
\,\!</math></center>
Such an identity matrix always has ones along the diagonal and zeroes everywhere else.  For example, the <math>3 \times 3\,\!</math> identity matrix is 
<center><math>
\mathbb{I} = \left(\begin{array}{ccc}
        1 & 0 & 0 \\
        0 & 1 & 0 \\
        0 & 0 & 1 \end{array}\right). 
\,\!</math></center>
It is straight-forward to verify that any <math>3 \times 3\,\!</math> matrix is not changed when multiplied by the identity matrix.

===Complex Conjugate===

The ''complex conjugate of a matrix'' <!-- \index{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: <math>A^*\,\!</math>.  For example,
{{Equation|<math>\begin{align}
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{align}
\,\!</math>|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'' <!-- \index{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 <math>3\times 3\,\!</math> matrix, the
transpose is given by
{{Equation|<math>\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}
\,\!</math>|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,
(<math>\dagger\,\!</math>):
{{Equation|<math>
(A^T)^* = (A^*)^T \equiv A^\dagger.
\,\!</math>|C.4}}
For our <math>3\times 3\,\!</math> example, 
<center><math>
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). 
\,\!</math></center>
If a matrix is its own Hermitian conjugate, i.e. <math>A^\dagger = A\,\!</math>, then
we call it a ''Hermitian matrix''.  <!-- \index{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.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)


===The Inverse of a Matrix===

===Index Notation===

Very often we write the product of two matrices <math>A\,\!</math> and <math>B\,\!</math> simply as
<math>AB\,\!</math> and let <math>C=AB\,\!</math>.  However, it is also quite useful to write this
in component form.  In this case, if these are <math>n\times n\,\!</math> matrices, the component form will be 
<center><math>
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.
\,\!</math></center>
This says that the element in the <math>i^{\mbox{th}}\,\!</math> row and
<math>j^{\mbox{th}}\,\!</math> column of the matrix <math>C\,\!</math> is the sum <math>\sum_{j=1}^n
a_{ij}b_{jk}\,\!</math>.  The transpose of <math>C\,\!</math> has elements
<center><math>
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.
\,\!</math></center>
Now if we were to transpose <math>A\,\!</math> and <math>B\,\!</math> as well, this would read
<center><math>
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_{j=1}^n b^T_{ij} a^T_{jk}.
\,\!</math></center>
This gives us a way of seeing the general rule that 
<center><math>
C^T = B^TA^T.  
\,\!</math></center>
It follows that 
<center><math>
C^\dagger = B^\dagger A^\dagger.
\,\!</math></center>

===The Trace===

The ''trace'' <!-- \index{trace}--> of a matrix is the sum of the diagonal
elements and is denoted <math>\mbox{Tr}\,\!</math>.  So for example, the trace of an
<math>n\times n\,\!</math> matrix <math>A\,\!</math> is 
<center><math>
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}
\,\!</math>.</center>

Some useful properties of the trace are the following:
#<math>\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!</math>
#<math>\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!</math>.
Using the first of these results,
<center><math>
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).
\,\!</math></center>
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 <math>n\times n\,\!</math> 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''<!--\index{determinant}--> of a <math>2\times 2\,\!</math> matrix, 
{{Equation|<math>
N = \left(\begin{array}{cc}
                 a & b \\
                 c & d \end{array}\right),
\,\!</math>|C.5}}
is given by
{{Equation|<math>
\det(N) = ad-bc.
\,\!</math>|C.6}}
Higher-order determinants can be written in terms of smaller ones in a recursive way.  For example, let 
<center>
<math>
M = \left(\begin{array}{ccc}
                 m_{11} & m_{12} & m_{13} \\
                 m_{21} & m_{22} & m_{23} \\
                 m_{31} & m_{32} & m_{33}\end{array}\right).
\,\!</math>
</center>  
Then 
<center>
<math>
\det(M) = m_{11}\det\left(\begin{array}{cc}
                 m_{22} & m_{23} \\
                 m_{32} & m_{33} \end{array}\right) 
- m_{12}\det\left(\begin{array}{cc}
                 m_{21} & m_{23} \\
                 m_{31} & m_{33} \end{array}\right)
+m_{13}\det\left(\begin{array}{cc}
                 m_{21} & m_{22} \\
                 m_{31} & m_{32} \end{array}\right).
\,\!</math>
</center>  



The ''determinant''<!-- \index{determinant}--> of a matrix <math>A\,\!</math> can be
also be written in terms of its components as 
{{Equation|<math>
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}
a_{1i}a_{2j}a_{3k}a_{4l} ...,
\,\!</math>|C.7}}
where the symbol 
{{Equation|<math>
\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}
\,\!</math>|C.8}}

Let us consider the example of the <math>3\times 3\,\!</math> matrix <math>A\,\!</math> given
above.  The determinant can be calculated by
<center><math>
\det(A) = \sum_{i,j,k} \epsilon_{ijk}
a_{1i}a_{2j}a_{3k},
\,\!</math></center>
where, explicitly, 
{{Equation|<math>
\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}
\,\!</math>|C.9}}
so that 
{{Equation|<math>\begin{align}
\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_{321}a_{13}a_{22}a_{31}.
\end{align}
\,\!</math>|C.10}}
Now given the values of <math>\epsilon_{ijk}\,\!</math> in [[#eqC.9|Eq. C.9]],
this is
<center><math>
\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_{22}a_{31}.
\,\!</math></center>

The determinant has several properties that are useful to know.  A few are listed here:  
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: <center><math> \det(A) = \det(A^T).\,\!</math></center>
#The determinant of a product is the product of determinants:    <center><math> \det(AB) = \det(A)\det(B).\,\!</math></center>
From this last property, another specific property can be derived.
If we take the determinant of the product of a matrix and its
inverse, we find
<center><math>
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,
\,\!</math></center>
since the determinant of the identity is one.  This implies that
<center><math>
\det(U^{-1}) = \frac{1}{\det(U)}.  
\,\!</math></center>

===The Inverse of a Matrix===


The inverse <!-- \index{inverse}--> of a square matrix <math>A\,\!</math> is another matrix,
denoted <math>A^{-1}\,\!</math>, such that 
<center><math>
AA^{-1} = A^{-1}A = \mathbb{I},
\,\!</math></center>
where <math>\mathbb{I}\,\!</math> is the identity matrix consisting of zeroes everywhere
except the diagonal, which has ones.  For example, the <math>3\times 3\,\!</math>
identity matrix is 
<center><math>
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1
  \end{array}\right). 
\,\!</math></center>

It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.''  Thus one only needs 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; primarily, it is because their eigenvalues are real.  Thus Hermitian matrices are used to represent density operators and density matrices, as well as Hamiltonians.  The density operator is a positive semi-definite Hermitian matrix (it has no negative eigenvalues) that has its trace equal to one.  In any case, it is often desirable to represent <math>N\times N\,\!</math> Hermitian matrices using a real linear combination of a complete set of <math>N\times N\,\!</math> Hermitian matrices.  A set of <math>N\times N\,\!</math> Hermitian matrices is complete if any Hermitian matrix can be represented in terms of the set.  Let <math>\{\lambda_i\}\,\!</math> be a complete set.  Then any Hermitian matrix can be represented by <math>\sum_i a_i \lambda_i\,\!</math>.  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 (its trace is one) because the identity part of an <math>N\times N\,\!</math> Hermitian matrix is <math>(1/N)\mathbb{I}\,\!</math> 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 where 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 - Qubits and Collections of Qubits|Chapter 2]] and in particular in [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|Section 2.4]].

===Unitary Matrices===

A ''unitary matrix'' <!-- \index{unitary matrix} --> <math>U\,\!</math> is one whose
inverse is also its Hermitian conjugate, <math>U^\dagger = U^{-1}\,\!</math>, so that 
<center><math>
U^\dagger U = UU^\dagger = \mathbb{I}.
\,\!</math></center>
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.<!-- \index{special unitary matrix}-->  The set of
<math>n\times n\,\!</math> unitary matrices is denoted
<math>U(n)\,\!</math> and the set of special unitary matrices is denoted <math>SU(n)\,\!</math>.  

Unitary matrices are particularly important in quantum mechanics
because they describe the evolution of quantum states.
They have this ability due to the fact that the rows and columns of unitary matrices (viewed as vectors) are orthonormal. (This is made clear in an example below.)  This means that when
they act on a basis vector of the form
{{Equation|<math> \left\vert j\right\rangle = 
 \left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 
  \end{array}\right), 
\,\!</math>|C.11}}
with a single 1, in say the <math>j</math>th spot, and zeroes everywhere else, the result is a normalized complex vector.  Acting on a set of
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]]
will produce another orthonormal set.  

Let us consider the example of a <math>2\times 2\,\!</math> unitary matrix, 
{{Equation|<math>
U = \left(\begin{array}{cc} 
              a & b \\ 
              c & d 
           \end{array}\right).
\,\!</math>|C.12}}
The inverse of this matrix is the Hermitian conjugate, 
{{Equation|<math>
U ^{-1} = U^\dagger = \left(\begin{array}{cc} 
                         a^* & c^* \\ 
                         b^* & d^* 
                       \end{array}\right),
\,\!</math>|C.13}}
provided that the matrix <math>U\,\!</math> satisfies the constraints
{{Equation|<math>\begin{align}
|a|^2 + |b|^2 = 1, \; & \; ac^*+bd^* =0  \\
ca^*+db^*=0,  \;      &  \; |c|^2 + |d|^2 =1,
\end{align}\,\!</math>|C.14}}
and
{{Equation|<math>\begin{align}
|a|^2 + |c|^2 = 1, \; & \; ba^*+dc^* =0  \\
b^*a+d^*c=0,  \;      &  \; |b|^2 + |d|^2 =1.
\end{align}\,\!</math>|C.15}}
Looking at each row as a vector, the constraints in
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the
vectors forming the rows.  Similarly, the constraints in
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the
vectors forming the columns.

===Inner and Outer Products===

Now that we have a definition for the Hermitian conjugate, we consider the
case for a <math>1\times n\,\!</math> matrix, i.e. a vector.  In Dirac notation, this is 
<center><math>
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta
  \end{array}\right).
\,\!</math></center>
The Hermitian conjugate comes up so often that we use the following
notation for vectors:
<center><math>
\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).  
\,\!</math></center>
This is a row vector and in Dirac notation is denoted by the symbol <math>\left\langle\cdot \right\vert\!</math>, which is called a ''bra''.  Let us consider a second complex vector, 
<center><math>
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta
  \end{array}\right). 
\,\!</math></center>
The ''inner product'' <!-- \index{inner product}--> between <math>\left\vert\psi\right\rangle\,\!</math> and 
<math>\left\vert\phi\right\rangle\,\!</math> is computed as follows:
{{Equation|<math>\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}
</math>|C.16}}

The ''outer product'' between these same two vectors is 
<center><math> 
\begin{align} (\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{align}\;\!</math></center>