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 $u:\mathbb {C} ^{n}\to \mathbb {C} ^{n}$ denote an arbitrary isometric linear map. Recall from the chapter on orthonormal matrices that any isometric map that maps ${\vec {0}}$ to ${\vec {0}}$ is linear.

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

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

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

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

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

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

Hermitian matrices

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

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

Quadratic forms

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

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

or as

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

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

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

Theorem

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

Proof

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

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