Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the $i$ -th row and $j$ -th column is equal to the complex conjugate of the element in the $j$ -th row and $i$ -th column, for all indices $i$ and $j$ : $A{\text{ is Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}$

For matrices with symmetry over the real number field, see Symmetric matrix.

or in matrix form: $A{\text{ is Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}.$

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

If the conjugate transpose of a matrix $A$ is denoted by $A^{\mathsf {H}},$ then the Hermitian property can be written concisely as

$A{\text{ is Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}$

Hermitian matrices are named after Charles Hermite,^[1] who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are $A^{\mathsf {H}}=A^{\dagger }=A^{\ast },$ although in quantum mechanics, $A^{\ast }$ typically means the complex conjugate only, and not the conjugate transpose.

Applications[edit]

Hermitian matrices are fundamental to quantum mechanics because they describe operators with necessarily real eigenvalues. An eigenvalue $a$ of an operator ${\hat {A}}$ on some quantum state $|\psi \rangle$ is one of the possible measurement outcomes of the operator, which requires the operators to have real eigenvalues.

In signal processing, Hermitian matrices are utilized in tasks like Fourier analysis and signal representation.^[2] The eigenvalues and eigenvectors of Hermitian matrices play a crucial role in analyzing signals and extracting meaningful information.

Hermitian matrices are extensively studied in linear algebra and numerical analysis. They have well-defined spectral properties, and many numerical algorithms, such as the Lanczos algorithm, exploit these properties for efficient computations. Hermitian matrices also appear in techniques like singular value decomposition (SVD) and eigenvalue decomposition.

In statistics and machine learning, Hermitian matrices are used in covariance matrices, where they represent the relationships between different variables. The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of multivariate distributions.^[3]

Hermitian matrices are applied in the design and analysis of communications system, especially in the field of multiple-input multiple-output (MIMO) systems. Channel matrices in MIMO systems often exhibit Hermitian properties.

In graph theory, Hermitian matrices are used to study the spectra of graphs. The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs.^[4] The Hermitian-adjacency matrix of a mixed graph is another important concept, as it is a Hermitian matrix that plays a role in studying the energies of mixed graphs.^[5]

Examples and solutions[edit]

In this section, the conjugate transpose of matrix $A$ is denoted as $A^{\mathsf {H}},$ the transpose of matrix $A$ is denoted as $A^{\mathsf {T}}$ and conjugate of matrix $A$ is denoted as ${\overline {A}}.$

See the following example:

${\begin{bmatrix}0&a-ib&c-id\\a+ib&1&m-in\\c+id&m+in&2\end{bmatrix}}$

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

Well-known families of Hermitian matrices include the Pauli matrices, the Gell-Mann matrices and their generalizations. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients,^[6]^[7] which results in skew-Hermitian matrices.

Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix $A$ equals the product of a matrix with its conjugate transpose, that is, $A=BB^{\mathsf {H}},$ then $A$ is a Hermitian positive semi-definite matrix. Furthermore, if $B$ is row full-rank, then $A$ is positive definite.

Properties[edit]

Main diagonal values are real[edit]

The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are real.

The sum of a square matrix and its conjugate transpose $\left(A+A^{\mathsf {H}}\right)$ is Hermitian.

The difference of a square matrix and its conjugate transpose $\left(A-A^{\mathsf {H}}\right)$ is (also called antihermitian). This implies that the commutator of two Hermitian matrices is skew-Hermitian.

skew-Hermitian

An arbitrary square matrix $C$ can be written as the sum of a Hermitian matrix $A$ and a skew-Hermitian matrix $B$ . This is known as the Toeplitz decomposition of $C$ .^: 227 $C=A+B\quad {\text{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\text{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)$

[9]

Additional facts related to Hermitian matrices include:

– Matrix equal to its transpose

Complex symmetric matrix

– Counts positive, negative, and zero eigenvalues of a block partitioned Hermitian matrix

Haynsworth inertia additivity formula

– Generalization of a bilinear form

Hermitian form

– Matrix that commutes with its conjugate transpose

Normal matrix

– Characterizes the diagonal of a Hermitian matrix with given eigenvalues

Schur–Horn theorem

– Linear operator equal to its own adjoint

Self-adjoint operator

– Matrix whose conjugate transpose is its negative (additive inverse) (anti-Hermitian matrix)

Skew-Hermitian matrix

– Complex matrix whose conjugate transpose equals its inverse

Unitary matrix

– Algebraic structure in linear algebra

Vector space

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Hermitian matrix"

by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.

Visualizing Hermitian Matrix as An Ellipse with Dr. Geo

. MathPages.com.