Hermitian matrix (original) (raw)

Properties

2. 1. The diagonal elements of a Hermitian matrix are real.
3. 1. The complex conjugate of a Hermitian matrix is a Hermitian matrix.
4. 1. If A is a Hermitian matrix, and B is a complex matrix of same order as A, then B⁢A⁢B∗ is a Hermitian matrix.
5. 1. A matrix is symmetric if and only if it is real and Hermitian.
7. 1. Hermitian matrices are also called self-adjoint since if A is Hermitian, then in the usualinner product of ℂn, we have
      for all u,v∈ℂn.

Example

1. 1. For any n×m matrix A, the n×n matrix A⁢A∗ is Hermitian.
2. 1. For any square matrix A, the Hermitian part of A,12⁢(A+A∗) is Hermitian. See this page (http://planetmath.org/DirectSumOfHermitianAndSkewHermitianMatrices).

3. 1. [11+i1+2⁢i1+3⁢i1-i22+2⁢i2+3⁢i1-2⁢i2-2⁢i33+3⁢i1-3⁢i2-3⁢i3-3⁢i4]

The first two examples are also examples of normal matrices.

Notes

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