Noun
Hermitian matrix (plural Hermitian matrixes or Hermitian matrices)
(linear algebra) A square matrix A with complex entries that is equal to its own conjugate transpose, i.e., such that
A
=
A
†
.
{\displaystyle A=A^{\dagger }.}
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.
If an observable can be described by a Hermitian matrix
H
{\displaystyle H}
, then for a given state
⟨
A
⟩
{\displaystyle \langle A\rangle }
, the expectation value of the observable for that state is
⟨
A
|
H
|
A
⟩
{\displaystyle \langle A|H|A\rangle }
.
A Hermitian matrix is positive semidefinite if and only if all of its principal minors are nonnegative. Source: Internet
A Hermitian matrix is positive definite if all its eigenvalues are positive. Source: Internet
Indefinite A Hermitian matrix which is neither positive definite, negative definite, positive-semidefinite, nor negative-semidefinite is called indefinite. Source: Internet
Negative-definite, semidefinite and indefinite matrices A Hermitian matrix is negative-definite, negative-semidefinite, or positive-semidefinite if and only if all of its eigenvalues are negative, non-positive, or non-negative, respectively. Source: Internet