## Section6.1Hermitian 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{6.1.1} \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{6.1.2} \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{6.1.3} \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{6.1.4} \end{equation}

so, we must have

\begin{equation} a^* = a, b^* = c, d^* = d\text{,}\tag{6.1.5} \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{6.1.6} \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{6.1.7} \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 $|v\rangle\langle v|\text{.}$