# Hermitian matrix

In mathematics, a **Hermitian matrix** (or **self-adjoint matrix**) is a square matrix with complex entries that is equal to its own conjugate transpose—that is, the element in the Template:Mvar-th row and Template:Mvar-th column is equal to the complex conjugate of the element in the Template:Mvar-th row and Template:Mvar-th column, for all indices Template:Mvar and Template:Mvar:

Hermitian matrices can be understood as the complex extension of real symmetric matrices.

If the conjugate transpose of a matrix is denoted by , then the Hermitian property can be written concisely as

Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having eigenvalues always real.

## Examples

See the following example:

The diagonal elements must be real, as they must be their own complex conjugate.

Well-known families of Pauli matrices, Gell-Mann matrices and their generalizations are Hermitian. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients,^{[1]}^{[2]} which results in *skew-Hermitian* matrices (see below).

Here we offer another useful Hermitian matrix using an abstract example. If a square matrix equals the multiplication of a matrix and its conjugate transpose, that is, , then is a Hermitian positive semi-definite matrix. Furthermore, if is row full-rank, then is positive definite.

## Properties

- The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are necessarily real, because they have to be equal to their complex conjugate. A matrix that has only real entries is Hermitian if and only if it is a symmetric matrix, i.e., if it is symmetric with respect to the main diagonal. A real and symmetric matrix is simply a special case of a Hermitian matrix.

- Every Hermitian matrix is a normal matrix.

- The finite-dimensional spectral theorem says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix Template:Mvar are real, and that Template:Mvar has Template:Mvar linearly independent eigenvectors. Moreover, it is possible to find an orthonormal basis of
**C**^{n}consisting of Template:Mvar eigenvectors of Template:Mvar.

- The sum of any two Hermitian matrices is Hermitian, and the inverse of an invertible Hermitian matrix is Hermitian as well. However, the product of two Hermitian matrices Template:Mvar and Template:Mvar is Hermitian if and only if
*AB*=*BA*. Thus*A*^{n}is Hermitian if Template:Mvar is Hermitian and Template:Mvar is an integer.

- For an arbitrary complex valued vector Template:Mvar the product is real because of . This is especially important in quantum physics where hermitian matrices are operators that measure properties of a system e.g. total spin which have to be real.

- The Hermitian complex Template:Mvar-by-Template:Mvar matrices do not form a vector space over the complex numbers, since the identity matrix
*I*_{n}is Hermitian, but*i**I*_{n}is not. However the complex Hermitian matrices*do*form a vector space over the real numbers**R**. In the 2*n*^{2}-dimensional vector space of complex*n*×*n*matrices over**R**, the complex Hermitian matrices form a subspace of dimension*n*^{2}. If*E*_{jk}denotes the Template:Mvar-by-Template:Mvar matrix with a 1 in the*j*,*k*position and zeros elsewhere, a basis can be described as follows:

- for (Template:Mvar matrices)

- together with the set of matrices of the form
- and the matrices
- where denotes the complex number , known as the imaginary unit.

- If Template:Mvar orthonormal eigenvectors of a Hermitian matrix are chosen and written as the columns of the matrix Template:Mvar, then one eigendecomposition of Template:Mvar is where and therefore

## Further properties

{{safesubst:#invoke:anchor|main}}Additional facts related to Hermitian matrices include:

- The sum of a square matrix and its conjugate transpose is Hermitian.
- The difference of a square matrix and its conjugate transpose is skew-Hermitian (also called antihermitian). This implies that commutator of two Hermitian matrices is skew-Hermitian.
- An arbitrary square matrix Template:Mvar can be written as the sum of a Hermitian matrix Template:Mvar and a skew-Hermitian matrix Template:Mvar:

- The determinant of a Hermitian matrix is real:

- (Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)

## Rayleigh quotient

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## See also

- Skew-Hermitian matrix (anti-Hermitian matrix)
- Haynsworth inertia additivity formula
- Hermitian form
- Self-adjoint operator
- Unitary matrix

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ Physics 125 Course Notes at California Institute of Technology

## External links

- {{#invoke:citation/CS1|citation

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- Visualizing Hermitian Matrix as An Ellipse with Dr. Geo, by Chao-Kuei Hung from Shu-Te University, gives a more geometric explanation.
- Template:MathPages