Hermitian Matrices

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Section 5.1 Hermitian Matrices

There are two uses of the word Hermitian, one is to describe a type of operation–the Hermitian adjoint (a verb), the other is to describe a type of operator–a Hermitian matrix or Hermitian adjoint (a noun).

On an \(n\times m\) matrix, \(N\text{,}\) the Hermitian adjoint (often denoted with a dagger, \(\dagger\text{,}\) means the conjugate transpose

\begin{equation} M^\dagger=M^*{}^T\tag{5.1.1} \end{equation}

A complex \(n\times n\) (square) matrix \(M\) is Hermitian if it equals its conjugate transpose, that is, if

\begin{equation} M^\dagger = M\text{.}\tag{5.1.2} \end{equation}

For example, let \(M\) be a \(2\times2\) complex matrix, so that

\begin{equation} M = \begin{pmatrix} a\amp b\\ c\amp d \end{pmatrix}\tag{5.1.3} \end{equation}

with \(a,b,c,d\in\CC\text{.}\) If \(M\) is Hermitian, then \(M^\dagger=M\text{.}\) But

\begin{equation} M^\dagger = \begin{pmatrix} a^*\amp c^*\\ b^*\amp d^* \end{pmatrix}\text{,}\tag{5.1.4} \end{equation}

so, we must have

\begin{equation} a^* = a, b^* = c, d^* = d\text{,}\tag{5.1.5} \end{equation}

i.e. \(a\) and \(d\) are real and \(c\) is the complex conjugate of \(b\text{.}\)

In index notation, if the components of \(M\) are denoted \(m_{ij}\text{,}\) then \(M\) is Hermitian if and only if

\begin{equation} m_{ij} = m^*_{ji}\tag{5.1.6} \end{equation}

for all \(i\text{,}\) \(j\text{.}\) Thus, the diagonal elements of a Hermitian matrix must be real, and the off-diagonal elements come in complex conjugate pairs, paired symmetrically across the main diagonal.

If \(M\) is both Hermitian and real, then \(M\) is a symmetric matrix. An anti-Hermitian matrix is one for which the Hermitian adjoint is the negative of the matrix:

\begin{equation} M^\dagger = -M\text{.}\tag{5.1.7} \end{equation}

An matrix which is both anti-Hermitian and real is antisymmetric.

An important special case of a Hermitian matrix can be constructed from any column vector \(v\) by computing its outer square, which in traditional vector notation would be written \(vv^\dagger\) and in bra/ket notation would be written \(|v\rangle\langle v|\text{.}\)