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

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

The standard basis vectors for ${ℂ}^n$, ${\overrightarrow{e}}_1,{\overrightarrow{e}}_2,\dots ,{\overrightarrow{e}}_n$, are all of unit length and are all mutually orthogonal: ${\overrightarrow{e}}_i\cdot {\overrightarrow{e}}_j=\{\begin{array}{cc}1& (i=j)\\ 0& (i\ne j)\end{array}$

If $U=\text{Rep}(u)=\left(\begin{array}{cccc}{\overrightarrow{u}}_1& {\overrightarrow{u}}_2& \dots & {\overrightarrow{u}}_n\end{array}\right)$ is the matrix that describes the isometric linear map $u$, then the columns ${\overrightarrow{u}}_i=u({\overrightarrow{e}}_i)$ are also all of unit length and are all mutually orthogonal: ${\overrightarrow{u}}_i\cdot {\overrightarrow{u}}_j=\{\begin{array}{cc}1& (i=j)\\ 0& (i\ne j)\end{array}$

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

${\left(\begin{array}{cccc}{a}_{1,1}& a_{1,2}& \dots & a_{1,m}\\ a_{2,1}& a_{2,2}& \dots & a_{2,m}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n,1}& a_{n,2}& \dots & a_{n,m}\end{array}\right)}^H=\left(\begin{array}{cccc}{a}_{1,1}^{*}& a_{2,1}^{*}& \dots & a_{n,1}^{*}\\ a_{1,2}^{*}& a_{2,2}^{*}& \dots & a_{n,2}^{*}\\ ⋮& ⋮& \ddots & ⋮\\ a_{1,m}^{*}& a_{2,m}^{*}& \dots & a_{n,m}^{*}\end{array}\right)$

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 $U^{H}U=U{U}^H=I$ (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, $A=\left(\begin{array}{cc}4& 6i\\ 6i& 5\end{array}\right)$ is symmetric but not Hermitian, but $A=\left(\begin{array}{cc}4& 6i\\ -6i& 5\end{array}\right)$ is Hermitian but not symmetric.

Quadratic forms

Given a square $n\times n$ matrix $A$ with real valued entries, the function $Q(\overrightarrow{x})={\overrightarrow{x}}^{T}A\overrightarrow{x}$ is a quadratic function over the entries of $\overrightarrow{x}\in {ℝ}^n$, 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:

$Q\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{cc}{x}_1& x_2\end{array}\right)\left(\begin{array}{cc}4& -7\\ 0& 5\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$

or as

$Q\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{cc}{x}_1& x_2\end{array}\right)\left(\begin{array}{cc}4& -3.5\\ -3.5& 5\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$

The coefficient of the term $x_i{x}_j$ for $i\ne j$ 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 $Q(\overrightarrow{x})={\overrightarrow{x}}^{H}A\overrightarrow{x}$, where $\overrightarrow{x}\in {ℂ}^n$ is arbitrary. Requiring that $A$ be Hermitian is similar to the requirement that $A$ be symmetric in the case of real numbers. $Q(\overrightarrow{x})$ always returns a real number if $A$ is Hermitian:

Template:BookCat