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
A complex \(n\times n\) (square) matrix \(M\) is Hermitian if it equals its conjugate transpose, that is, if
For example, let \(M\) be a \(2\times2\) complex matrix, so that
with \(a,b,c,d\in\CC\text{.}\) If \(M\) is Hermitian, then \(M^\dagger=M\text{.}\) But
so, we must have
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
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:
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 \(|v\rangle\langle v|\text{.}\)