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Hermitian matrix (Definition)

For a complex matrix $ A$, let $ A^\ast=\overline{A}^{T}$, where $ A^T$ is the transpose, and $ \bar{A}$ is the complex conjugate of $ A$.

Definition A complex square matrix $ A$ is Hermitian, if

$\displaystyle A = A^*. $

Properties

  1. The eigenvalues of a Hermitian matrix are real.
  2. The diagonal elements of a Hermitian matrix are real.
  3. The complex conjugate of a Hermitian matrix is a Hermitian matrix.
  4. If $ A$ is a Hermitian matrix, and $ B$ is a complex matrix of same order as $ A$, then $ BAB^\ast$ is a Hermitian matrix.
  5. A matrix is symmetric if and only if it is real and Hermitian.
  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.
  7. Hermitian matrices are also called self-adjoint since if $ A$ is Hermitian, then in the usual inner product of $ \mathbb{C}^n$, we have
    $\displaystyle \langle u,Av \rangle = \langle Au,v\rangle$
    for all $ u,v\in \mathbb{C}^n$.

Example

  1. For any $ n\times m$ matrix $ A$, the $ n\times n$ matrix $ A A^\ast$ is Hermitian.
  2. For any square matrix $ A$, the Hermitian part of $ A$, $ \frac{1}{2}(A+A^\ast)$ is Hermitian. See this page.
  3. $\displaystyle \begin{bmatrix} 1 & 1 + i & 1 + 2i & 1 + 3i \ 1 - i & 2 & 2 + 2... ... 1 - 2i & 2 - 2i & 3 & 3 + 3i \ 1 - 3i & 2 - 3i & 3 - 3i & 4 \end{bmatrix} $
The first two examples are also examples of normal matrices.

Notes

Hermitian matrices are named after Charles Hermite (1822-1901) [2], who proved in 1855 that the eigenvalues of these matrices are always real [1].

Bibliography

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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See Also: self-dual, skew-Hermitian matrix, self-adjoint operator, Pauli matrices

Other names:  Hermitian, self-adjoint

Attachments:
direct sum of Hermitian and skew-Hermitian matrices (Example) by mathcam
eigenvalues of a Hermitian matrix are real (Theorem) by Andrea Ambrosio
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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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