## Section5.1Hermitian 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{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 $$|v\rangle\langle v|\text{.}$$