Hermitian matrices have real diagonal elements as well as real eigenvalues.
If a Hermitian matrix has a simple spectrum (of eigenvalues) then its eigenvectors are orthogonal. On the other hand, a set of two or more eigenvectors with the same eigenvalue can be orthogonalized (e.g., through the Gram–Schmidt process, since any linear combination of equal-eigenvalue eigenvectors will also be an eigenvect) and will already be orthogonal to other eigenvectors which have different eigenvalues.
If an observable can be described by a Hermitian matrix , then for a given state , the expectation value of the observable for that state is .
Origin of hermitian-matrix
- Named after Charles Hermite (1822–1901), French mathematician.