Hermitian Matrix

Hermitian matrix

[er′mish·ən ′mā·triks]

(mathematics)

A matrix which equals its conjugate transpose matrix, that is, is self-adjoint.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Hermitian Matrix

 

(or self-adjoint matrix), a matrix coincident with its adjoint, that is, a matrix such that a_ik= ā_ki, where ā is the complex conjugate of the number a. If the elements of a Hermitian matrix are real, then the matrix is symmetric. A Hermitian matrix has real eigenvalues λ₁, λ₂, …, λ_n and corresponds to a linear transformation in a complex n-dimensional space that reduces to stretchings by ǀλ_iǀ in n mutually perpendicular directions and reflections in the planes orthogonal to the directions for which λ_i < 0. A bilinear form

whose coefficients form a Hermitian matrix, is called a Hermitian form. Any matrix can be written in the form A₁ + iA₂, where A₁ and A₂ are Hermitian matrices, and in the form A ∪, where A is a Hermitian matrix and U is a unitary matrix. If A and B are Hermitian matrices, then A B is a Hermitian matrix if and only if A and B commute.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.