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Hermitian matrix
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(Definition)
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For a complex matrix , let
, where is the transpose, and is the complex conjugate of .
Definition A complex square matrix is Hermitian, if
- The eigenvalues of a Hermitian matrix are real.
- The diagonal elements of a Hermitian matrix are real.
- The complex conjugate of a Hermitian matrix is a Hermitian matrix.
- If
is a Hermitian matrix, and is a complex matrix of same order as , then is a Hermitian matrix.
- A matrix is symmetric if and only if it is real and Hermitian.
- Hermitian matrices are a vector subspace of the vector space of complex matrices. The real symmetric matrices are a subspace of the Hermitian matrices.
- Hermitian matrices are also called self-adjoint since if
is Hermitian, then in the usual inner product of
, we have
for all
.
- For any
matrix , the matrix is Hermitian.
- For any square matrix
, the Hermitian part of ,
is Hermitian. See this page.
-
The first two examples are also examples of normal matrices.
Hermitian matrices are named after Charles Hermite (1822-1901) [2], who proved in 1855 that the eigenvalues of these matrices are always real [1].
- 1
- H. Eves, Elementary Matrix Theory, Dover publications, 1980.
- 2
- The MacTutor History of Mathematics archive, Charles Hermite
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"Hermitian matrix" is owned by matte. [ full author list (5) | owner history (1) ]
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Cross-references: eigenvalues, normal matrices, inner product, symmetric matrices, vector space, vector subspace, symmetric, order, real, diagonal, eigenvalues of a Hermitian matrix are real, square matrix, complex conjugate, transpose, matrix, complex
There are 38 references to this entry.
This is version 14 of Hermitian matrix, born on 2002-01-21, modified 2008-03-28.
Object id is 1505, canonical name is HermitianMatrix.
Accessed 31380 times total.
Classification:
| AMS MSC: | 15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices ) |
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Pending Errata and Addenda
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