Hermitian matrix
For a complex matrix , let , where is the transpose, and is the complex conjugate of .
Definition A complex square matrix is Hermitian, if
Properties
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1.
The eigenvalues of a Hermitian matrix are real.
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2.
The diagonal elements of a Hermitian matrix are real.
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3.
The complex conjugate of a Hermitian matrix is a Hermitian matrix.
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4.
If is a Hermitian matrix, and is a complex matrix of same order as , then is a Hermitian matrix.
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5.
A matrix is symmetric if and only if it is real and Hermitian.
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6.
Hermitian matrices are a vector subspace of the vector space of complex matrices. The real symmetric matrices are a subspace of the Hermitian matrices.
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7.
Hermitian matrices are also called self-adjoint since if is Hermitian, then in the usual inner product of , we have
for all .
Example
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1.
For any matrix , the matrix is Hermitian.
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2.
For any square matrix , the Hermitian part of , is Hermitian. See this page (http://planetmath.org/DirectSumOfHermitianAndSkewHermitianMatrices).
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3.
The first two examples are also examples of normal matrices.
Notes
- 1.
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2.
Hermitian, or self-adjoint operators on a Hilbert space play a fundamental role in quantum theories as their eigenvalues are observable, or measurable; such Hermitian operators can be represented by Hermitian matrices.
References
- 1 H. Eves, Elementary Matrix Theory, Dover publications, 1980.
- 2 The MacTutor History of Mathematics archive, http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Hermite.htmlCharles Hermite
Title | Hermitian matrix |
Canonical name | HermitianMatrix |
Date of creation | 2013-03-22 12:12:00 |
Last modified on | 2013-03-22 12:12:00 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 21 |
Author | matte (1858) |
Entry type | Definition |
Classification | msc 15A57 |
Synonym | Hermitian |
Synonym | self-adjoint |
Related topic | SelfDual |
Related topic | SkewHermitianMatrix |
Related topic | SelfAdjointOperator |
Related topic | PauliMatrices |
Defines | Hermitian operator |