Hermitian matrix

English

Wikipedia

Etymology

Named after Charles Hermite (1822–1901), French mathematician.

Pronunciation

• "her mission matrix"
• (US) IPA: /hɝ.ˈmɪ.ʃən ˈmeɪ.tɹɪks/

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 $A = A^\dagger\,,$ where $A^\dagger$ 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 eigenvector) and will already be orthogonal to other eigenvectors which have different eigenvalues.

   If an observable can be described by a Hermitian matrix $H$, then for a given state $|A\rangle$, the expectation value of the observable for that state is $\langle A|H|A\rangle$.

Synonyms

• (hermitian matrix): self-adjoint matrix

Hyponyms

• Pauli matrix

Translations

mathematics: square matrix that is equal to its own conjugate transpose

• Icelandic: sjálfoka fylki (is) n, hermískt fylki (is) n

References