English

Noun

Hermitian matrix ‎(plural Hermitian matrices)

1. (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.^[1]

   If a Hermitian matrix has a simple spectrum (of eigenvalues) then its eigenvectors are orthogonal.^[2]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 eigenvector) 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 .

Synonyms

• (hermitian matrix): self-adjoint matrix

Hyponyms

• Pauli matrix

Translations

References

1. ↑ Proof Wiki — Hermitian Operators have Real Eigenvalues
2. ↑ Proof Wiki — Hermitian Operators have Orthogonal Eigenvectors