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, $N$, the Hermitian adjoint (often denoted with a dagger, $\dagger$, means the conjugate transpose \begin{equation} M^\dagger=M^*{}^T \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 . \end{equation}

For example, let $M$ be a $2\times2$ complex matrix, so that \begin{equation} M = \begin{pmatrix} a& b\\ c& d\\ \end{pmatrix} \end{equation} with $a,b,c,d\in\CC$. If $M$ is Hermitian, then $M^\dagger=M$. But \begin{equation} M^\dagger = \begin{pmatrix} a^*& c^*\\ b^*& d^*\\ \end{pmatrix} , \end{equation} so, we must have \begin{equation} a^* = a, \quad b^* = c, \quad d^* = d , \end{equation} i.e. $a$ and $d$ are real and $c$ is the complex conjugate of $b$.

In index notation, if the components of $M$ are denoted $m_{ij}$, then $M$ is Hermitian if and only if \begin{equation} m_{ij} = m^*_{ji} \end{equation} for all $i$, $j$. 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 . \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 $|v\rangle\langle v|$.

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