# Hermitian Matrix

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## Hermitian matrix

[er′mish·ən ′mā·triks]## 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 λ_{1}, λ_{2}, …, λ_{n} and corresponds to a linear transformation in a complex n-dimensional space that reduces to stretchings by ǀλ_{i}ǀ 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 *A _{1} + iA_{2}*, where

*A*

_{1}and

*A*

_{2}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.