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

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

The standard basis vectors for {ℂ}^{n}, {\overset{→}{e}}_{1} , {\overset{→}{e}}_{2} , … , {\overset{→}{e}}_{n}, are all of unit length and are all mutually orthogonal: {\overset{→}{e}}_{i} ⋅ {\overset{→}{e}}_{j} = \{ \begin{matrix}1 & ( i = j ) \\ 0 & ( i ≠ j )\end{matrix}

If U = \text{Rep} ( u ) = ( \begin{matrix}{\overset{→}{u}}_{1} & {\overset{→}{u}}_{2} & … & {\overset{→}{u}}_{n}\end{matrix} ) is the matrix that describes the isometric linear map u, then the columns {\overset{→}{u}}_{i} = u ( {\overset{→}{e}}_{i} ) are also all of unit length and are all mutually orthogonal:
{\overset{→}{u}}_{i} ⋅ {\overset{→}{u}}_{j} = \{ \begin{matrix}1 & ( i = j ) \\ 0 & ( i ≠ j )\end{matrix}

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

{( \begin{matrix}{a}_{1 , 1} & {a}_{1 , 2} & … & {a}_{1 , m} \\ {a}_{2 , 1} & {a}_{2 , 2} & … & {a}_{2 , m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {a}_{n , 1} & {a}_{n , 2} & … & {a}_{n , m}\end{matrix} )}^{H} = ( \begin{matrix}{a}_{1 , 1}^{∗} & {a}_{2 , 1}^{∗} & … & {a}_{n , 1}^{∗} \\ {a}_{1 , 2}^{∗} & {a}_{2 , 2}^{∗} & … & {a}_{n , 2}^{∗} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {a}_{1 , m}^{∗} & {a}_{2 , m}^{∗} & … & {a}_{n , m}^{∗}\end{matrix} )

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 × 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{matrix}4 & 6 i \\ 6 i & 5\end{matrix} ) is symmetric but not Hermitian, but A = ( \begin{matrix}4 & 6 i \\ − 6 i & 5\end{matrix} ) is Hermitian but not symmetric.

Quadratic forms

Given a square n × n matrix A with real valued entries, the function Q ( \overset{→}{x} ) = {\overset{→}{x}}^{T} A \overset{→}{x} is a quadratic function over the entries of \overset{→}{x} ∈ {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} ) = 4 {x}_{1}^{2} − 7 {x}_{1} {x}_{2} + 5 {x}_{2}^{2}, Q ( {x}_{1} , {x}_{2} ) can be expressed as:

Q ( \begin{matrix}{x}_{1} \\ {x}_{2}\end{matrix} ) = ( \begin{matrix}{x}_{1} & {x}_{2}\end{matrix} ) ( \begin{matrix}4 & − 7 \\ 0 & 5\end{matrix} ) ( \begin{matrix}{x}_{1} \\ {x}_{2}\end{matrix} )

or as

Q ( \begin{matrix}{x}_{1} \\ {x}_{2}\end{matrix} ) = ( \begin{matrix}{x}_{1} & {x}_{2}\end{matrix} ) ( \begin{matrix}4 & − 3.5 \\ − 3.5 & 5\end{matrix} ) ( \begin{matrix}{x}_{1} \\ {x}_{2}\end{matrix} )

The coefficient of the term {x}_{i} {x}_{j} for i ≠ 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 ( \overset{→}{x} ) = {\overset{→}{x}}^{H} A \overset{→}{x}, where \overset{→}{x} ∈ {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 ( \overset{→}{x} ) always returns a real number if A is Hermitian:

Theorem

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

Proof

∀ \overset{→}{x} ∈ {ℂ}^{n} : {\overset{→}{x}}^{H} A \overset{→}{x} ∈ ℝ \ ⟺ \ ∀ \overset{→}{x} ∈ {ℂ}^{n} : {\overset{→}{x}}^{H} A \overset{→}{x} = ( {\overset{→}{x}}^{H} A \overset{→}{x} {)}^{∗} \ ⟺ \ ∀ \overset{→}{x} ∈ {ℂ}^{n} : {\overset{→}{x}}^{H} A \overset{→}{x} = {\overset{→}{x}}^{H} {A}^{H} \overset{→}{x}

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

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