Hermitian-matrix Definition

noun

(linear algebra) A square matrix with complex entries that is equal to its own conjugate transpose, i.e., a matrix such that where denotes the conjugate transpose of a matrix A.

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 .

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