Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a  that is equal to its own —that is, the element in the $i$-th row and $j$-th column is equal to the  of the element in the $j$-th row and $i$-th column, for all indices $i$ and $j$:

or in matrix form:


 * $$A \text{ Hermitian} \quad \iff \quad A = \overline {A^\mathsf{T}}$$.

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

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

Hermitian matrices are named after, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real. Other, equivalent notations in common use are $$A^\mathsf{H} = A^\dagger = A^\ast$$, although note that in, $$A^\ast$$ typically means the only, and not the.

Alternative characterizations
Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:

Equality with the adjoint
A square matrix $$A$$ is Hermitian if and only if it is equal to its, that is, it satisfies $$\langle w, Av\rangle = \langle Aw, v\rangle,$$ for any pair of vectors $$v, w$$, where $$\langle \cdot, \cdot\rangle$$ denotes operation.

This is also the way that the more general concept of is defined.

Reality of quadratic forms
A square matrix $$A$$ is Hermitian if and only if it is such that $$\langle v,Av\rangle\in\mathbb R, \quad v\in V.$$

Spectral properties
A square matrix $$A$$ is Hermitian if and only if it is unitarily with real.

Applications
Hermitian matrices are fundamental to the quantum theory of created by, , and  in 1925.

Examples
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}

2    & 2 + i & 4 \\ 2 - i & 3    & i \\ 4    &   - i & 1 \\ \end{bmatrix}$$

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

Well-known families of, and their generalizations are Hermitian. In such Hermitian matrices are often multiplied by  coefficients, which results in skew-Hermitian matrices (see ).

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

Properties

 * The entries on the (top left to bottom right) of any Hermitian matrix are.
 * Proof: By definition of the Hermitian matrix
 * $$H_{ij} = \overline{H}_{ji} $$
 * so for $i = j$ the above follows.
 * Only the entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in their s, as long as diagonally-opposite entries are complex conjugates.


 * A matrix that has only real entries is Hermitian it is . A real and symmetric matrix is simply a special case of a Hermitian matrix.
 * Proof: $$H_{ij} = \overline{H}_{ji}$$ by definition. Thus $H_{ij} = H_{ji}$ (matrix symmetry) if and only if $$H_{ij} = \overline{H}_{ij}$$ ($H_{ij}$ is real).


 * Every Hermitian matrix is a . That is to say, $AA^{H} = A^{H}A$.
 * Proof: $A = A^{H}$, so $AA^{H} = AA = A^{H}A$.


 * The finite-dimensional says that any Hermitian matrix can be  by a, and that the resulting diagonal matrix has only real entries. This implies that all s of a Hermitian matrix $A$ with dimension $n$ are real, and that $A$ has $n$ linearly independent s. Moreover, a Hermitian matrix has  eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an  of $ℂ^{n}$ consisting of $n$ eigenvectors of $A$.


 * The sum of any two Hermitian matrices is Hermitian.
 * Proof: $$(A + B)_{ij} = A_{ij} + B_{ij} = \overline{A}_{ji} + \overline{B}_{ji} = \overline{(A + B)}_{ji},$$ as claimed.


 * The of an invertible Hermitian matrix is Hermitian as well.
 * Proof: If $$A^{-1}A=I$$, then $$I= I^H = (A^{-1}A)^H=A^H(A^{-1})^H=A (A^{-1})^H$$, so $$A^{-1}=(A^{-1})^H$$ as claimed.


 * The of two Hermitian matrices $A$ and $B$ is Hermitian if and only if $AB = BA$.
 * Proof: Note that $$(AB)^\mathsf{H} = \overline{(AB)^\mathsf{T}} = \overline{B^\mathsf{T} A^\mathsf{T}} = \overline{B^\mathsf{T}} \overline{A^\mathsf{T}} = B^\mathsf{H} A^\mathsf{H} = BA.$$ Thus $$(AB)^\mathsf{H} = AB$$ $$AB = BA$$.
 * Thus $A^{n}$ is Hermitian if $A$ is Hermitian and $n$ is an integer.


 * For an arbitrary complex valued vector $v$ the product $$ v^\mathsf{H} A v $$ is real because of $$ v^\mathsf{H} A v = \left(v^\mathsf{H} A v\right)^\mathsf{H} $$. This is especially important in quantum physics where Hermitian matrices are operators that measure properties of a system e.g. total which have to be real.


 * The Hermitian complex $n$-by-$n$ matrices do not form a over the s, $ℂ$, 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  $ℝ$. In the $2n^{2}$- 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 $n$-by-$n$ matrix with a $1$ in the $j,k$ position and zeros elsewhere, a basis (orthonormal w.r.t. the Frobenius inner product) can be described as follows:


 * $$E_{jj} \text{ for } 1 \leq j \leq n \quad (n \text{ matrices}) $$


 * together with the set of matrices of the form


 * $$\frac{1}{\sqrt{2}}\left(E_{jk} + E_{kj}\right) \text{ for } 1 \leq j < k \leq n \quad \left( \frac{n^2-n} 2 \text{ matrices} \right) $$


 * and the matrices


 * $$\frac{i}{\sqrt{2}}\left(E_{jk} - E_{kj}\right) \text{ for } 1 \leq j < k \leq n \quad \left( \frac{n^2-n} 2 \text{ matrices} \right) $$


 * where $$i$$ denotes the complex number $$\sqrt{-1}$$, called the .


 * If $n$ orthonormal eigenvectors $$u_1, \dots, u_n$$ of a Hermitian matrix are chosen and written as the columns of the matrix $U$, then one of $A$ is $$ A = U \Lambda U^\mathsf{H}$$ where $$U U^\mathsf{H} = I = U^\mathsf{H} U$$ and therefore
 * $$A = \sum_j \lambda_j u_j u_j ^\mathsf{H},$$
 * where $$\lambda_j$$ are the eigenvalues on the diagonal of the diagonal matrix $$\; \Lambda $$.


 * The determinant of a Hermitian matrix is real:
 * Proof: $$ \det(A) = \det\left(A^\mathsf{T}\right)\quad \Rightarrow \quad \det\left(A^\mathsf{H}\right) = \overline{\det(A)} $$
 * Therefore if $$A = A^\mathsf{H}\quad \Rightarrow \quad \det(A) = \overline{\det(A)} $$.
 * (Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)

Decomposition into Hermitian and skew-Hermitian
Additional facts related to Hermitian matrices include:
 * 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  of two Hermitian matrices is 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 $C$.
 * $$C = A + B \quad\mbox{with}\quad A = \frac{1}{2}\left(C + C^\mathsf{H}\right) \quad\mbox{and}\quad B = \frac{1}{2}\left(C - C^\mathsf{H}\right)$$

Rayleigh quotient
In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient $$R(M, x)$$, is defined as:
 * $$R(M, x) := \frac{x^\mathsf{H} M x}{x^\mathsf{H} x}$$.

For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose $$x^\mathsf{H}$$ to the usual transpose $$x^\mathsf{T}$$. Note that $$R(M, c x) = R(M, x)$$ for any non-zero real scalar $$c$$. Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.

It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value $$\lambda_\min$$ (the smallest eigenvalue of M) when $$x$$ is $$v_\min$$ (the corresponding eigenvector). Similarly, $$R(M, x) \leq \lambda_\max$$ and $$R(M, v_\max) = \lambda_\max$$.

The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.

The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, $$\lambda_\max$$ is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to $M$ associates the Rayleigh quotient $R(M, x)$ for a fixed $x$ and $M$ varying through the algebra would be referred to as "vector state" of the algebra.