Linear Algebra/Unitary and Hermitian matrices

Unitary Matrices

Of considerable interest are linear maps that are "isometric", also known as "distance preserving maps". Such a map is also called an "isometry". Let ${\displaystyle u:\mathbb {C} ^{n}\to \mathbb {C} ^{n}}$  denote an arbitrary isometric linear map. Recall from the chapter on orthonormal matrices that any isometric map that maps ${\displaystyle {\vec {0}}}$  to ${\displaystyle {\vec {0}}}$  is linear.

The distance preserving nature of isometries also means that angles are preserved. If ${\displaystyle {\vec {a}},{\vec {b}}\in \mathbb {C} ^{n}}$  are arbitrary vectors, then the dot product is preserved by isometric transformations: ${\displaystyle u({\vec {a}})\cdot u({\vec {b}})={\vec {a}}\cdot {\vec {b}}}$ .

The standard basis vectors for ${\displaystyle \mathbb {C} ^{n}}$ , ${\displaystyle {\vec {e}}_{1},{\vec {e}}_{2},\dots ,{\vec {e}}_{n}}$ , are all of unit length and are all mutually orthogonal: ${\displaystyle {\vec {e}}_{i}\cdot {\vec {e}}_{j}=\left\{{\begin{array}{cc}1&(i=j)\\0&(i\neq j)\end{array}}\right.}$

If ${\displaystyle U={\text{Rep}}(u)={\begin{pmatrix}{\vec {u}}_{1}&{\vec {u}}_{2}&\dots &{\vec {u}}_{n}\\\end{pmatrix}}}$  is the matrix that describes the isometric linear map ${\displaystyle u}$ , then the columns ${\displaystyle {\vec {u}}_{i}=u({\vec {e}}_{i})}$  are also all of unit length and are all mutually orthogonal: ${\displaystyle {\vec {u}}_{i}\cdot {\vec {u}}_{j}=\left\{{\begin{array}{cc}1&(i=j)\\0&(i\neq j)\end{array}}\right.}$

The "Hermitian Transpose" of a matrix is the transpose with the conjugation of complex numbers applied on top:

${\displaystyle {\begin{pmatrix}a_{1,1}&a_{1,2}&\dots &a_{1,m}\\a_{2,1}&a_{2,2}&\dots &a_{2,m}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\dots &a_{n,m}\\\end{pmatrix}}^{H}={\begin{pmatrix}a_{1,1}^{*}&a_{2,1}^{*}&\dots &a_{n,1}^{*}\\a_{1,2}^{*}&a_{2,2}^{*}&\dots &a_{n,2}^{*}\\\vdots &\vdots &\ddots &\vdots \\a_{1,m}^{*}&a_{2,m}^{*}&\dots &a_{n,m}^{*}\\\end{pmatrix}}}$

The orthonormal properties of the columns of ${\displaystyle U}$  imply that the inverse of ${\displaystyle U}$  is simply its Hermitian transpose: ${\displaystyle U^{-1}=U^{H}}$ . Any matrix whose inverse is its Hermitian transpose is referred to as being "unitary". The key property of a unitary matrix ${\displaystyle U}$  is that ${\displaystyle U}$  be square and that ${\displaystyle U^{H}U=UU^{H}=I}$  (note that ${\displaystyle I={\text{Rep}}({\text{id}})}$  is the identity matrix). Unitary matrices denote isometric linear maps.

Hermitian matrices

Given a square ${\displaystyle n\times n}$  matrix ${\displaystyle A}$ , analogous to how ${\displaystyle A}$  is symmetric if ${\displaystyle A^{T}=A}$ , ${\displaystyle A}$  is Hermitian if ${\displaystyle A^{H}=A}$ , meaning that diagonally opposite entries of ${\displaystyle A}$  are complex conjugates of each other.

For example, ${\displaystyle A={\begin{pmatrix}4&6i\\6i&5\end{pmatrix}}}$  is symmetric but not Hermitian, but ${\displaystyle A={\begin{pmatrix}4&6i\\-6i&5\end{pmatrix}}}$  is Hermitian but not symmetric.

Given a square ${\displaystyle n\times n}$  matrix ${\displaystyle A}$  with real valued entries, the function ${\displaystyle Q({\vec {x}})={\vec {x}}^{T}A{\vec {x}}}$  is a quadratic function over the entries of ${\displaystyle {\vec {x}}\in \mathbb {R} ^{n}}$ , referred to as a "quadratic form". All terms in a quadratic form have degree 2. For instance, given the quadratic form ${\displaystyle Q(x_{1},x_{2})=4x_{1}^{2}-7x_{1}x_{2}+5x_{2}^{2}}$ , ${\displaystyle Q(x_{1},x_{2})}$  can be expressed as:

${\displaystyle Q{\begin{pmatrix}x_{1}\\x_{2}\\\end{pmatrix}}={\begin{pmatrix}x_{1}&x_{2}\\\end{pmatrix}}{\begin{pmatrix}4&-7\\0&5\\\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\\end{pmatrix}}}$

or as

${\displaystyle Q{\begin{pmatrix}x_{1}\\x_{2}\\\end{pmatrix}}={\begin{pmatrix}x_{1}&x_{2}\\\end{pmatrix}}{\begin{pmatrix}4&-3.5\\-3.5&5\\\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\\end{pmatrix}}}$

The coefficient of the term ${\displaystyle x_{i}x_{j}}$  for ${\displaystyle i\neq j}$  is the sum of the ${\displaystyle (i,j)}$  and ${\displaystyle (j,i)}$  entries. It then becomes sensible to split the coefficient of ${\displaystyle x_{i}x_{j}}$  between the ${\displaystyle (i,j)}$  and ${\displaystyle (j,i)}$  entries, in essence requiring ${\displaystyle A}$  to be symmetric: ${\displaystyle A^{T}=A}$ .

Generalizing to complex numbers, consider the quadratic form ${\displaystyle Q({\vec {x}})={\vec {x}}^{H}A{\vec {x}}}$ , where ${\displaystyle {\vec {x}}\in \mathbb {C} ^{n}}$  is arbitrary. Requiring that ${\displaystyle A}$  be Hermitian is similar to the requirement that ${\displaystyle A}$  be symmetric in the case of real numbers. ${\displaystyle Q({\vec {x}})}$  always returns a real number if ${\displaystyle A}$  is Hermitian:

Theorem

If ${\displaystyle A}$  is Hermitian, then the quadratic form ${\displaystyle Q({\vec {x}})={\vec {x}}^{H}A{\vec {x}}}$  always returns a real number.

Proof

${\displaystyle \forall {\vec {x}}\in \mathbb {C} ^{n}:{\vec {x}}^{H}A{\vec {x}}\in \mathbb {R} \iff \forall {\vec {x}}\in \mathbb {C} ^{n}:{\vec {x}}^{H}A{\vec {x}}=({\vec {x}}^{H}A{\vec {x}})^{*}\iff \forall {\vec {x}}\in \mathbb {C} ^{n}:{\vec {x}}^{H}A{\vec {x}}={\vec {x}}^{H}A^{H}{\vec {x}}}$

Since ${\displaystyle A^{H}=A}$  due to ${\displaystyle A}$  being Hermitian, ${\displaystyle Q({\vec {x}})}$  always returns a real number.