Section5.1Definition of Hermitian Matrices

On an $$n\times m$$ matrix, $$N\text{,}$$ you can take the Hermitian adjoint (usually denoted with a dagger, $$\dagger$$) which means take both the (complex) conjugate Section 2.2 and the transpose Section 3.4, in either order

$$N^\dagger=N^*{}^T\text{.}\tag{5.1.1}$$

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

$$M^\dagger = M\text{.}\tag{5.1.2}$$

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

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

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

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

so, we must have

$$a^* = a, b^* = c, d^* = d\text{,}\tag{5.1.5}$$

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

$$m_{ij} = m^*_{ji}\tag{5.1.6}$$

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 a matrix $$M$$ is both Hermitian and real, then $$M$$ is called a symmetric matrix.

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 $$|v\rangle\langle v|\text{.}$$

An anti-Hermitian matrix is one for which the Hermitian adjoint is the negative of the matrix:

$$M^\dagger = -M\text{.}\tag{5.1.7}$$

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