In mathematics, more precisely in linear algebra, a Hermitian matrix (or self-adjoint matrix) is a square matrix that is equal to its own conjugate transposethat is, its element in the -th row and -th column is the complex conjugate of its element in the -th row and -th column, for all indices and . With matrix notations:

:<math display="block">A \text{ is Hermitian} \quad \iff \quad A = \overline{A^\mathsf{T \quad \iff \quad A_{ij} = \overline{A_{ji,</math>

where <math>A^\mathsf{T}</math> is the transpose of <math>A, ~</math> <math>\overline{B}</math> is the conjugate of <math>B,</math> and <math>A_{ij}</math> is the element in the -th row and -th column of <math>A.</math>

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

The Hermitian property of a matrix <math>A</math> can be written concisely as

:<math display="block">A \text{ is Hermitian} \quad \iff \quad A = A^\mathsf{H},</math>

where <math>A^\mathsf{H}</math> denotes the conjugate transpose of <math>A.</math>

Equivalent notations in common use are <math>A^\mathsf{H} = A^\dagger = A^*,</math> although in quantum mechanics, <math>A^*</math> typically means the complex conjugate only, and not the conjugate transpose.

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

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 <math>A</math> is Hermitian if and only if it is equal to its conjugate transpose, that is, for any pair of vectors <math>\mathbf v, \mathbf w,</math> it satisfies

:<math display="block">\langle \mathbf v, A \mathbf w \rangle = \langle A \mathbf v, \mathbf w \rangle,</math>

where <math>\langle \cdot, \cdot \rangle</math> denotes the inner product operation.

This is also the way that the more general concept of self-adjoint operator is defined.

Real-valuedness of quadratic forms

An <math>n \times n</math> matrix <math>A</math> is Hermitian if and only if

:<math display="block">\langle \mathbf{v}, A \mathbf{v} \rangle \in \R, \quad \text{for all } \mathbf{v} \in \C^n.</math>

Spectral properties

A square matrix is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues.

Applications

Hermitian matrices are fundamental to quantum mechanics because they describe operators with necessarily real eigenvalues. An eigenvalue <math>a</math> of an operator <math>\hat{A}</math> on some quantum state <math>| \psi \rangle</math> 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. 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 as covariance matrices, which represent the relationships between different variables. The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of multivariate distributions.

Hermitian matrices are applied in the design and analysis of communications systems, 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. 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.

Examples

In the following example, <math>\mathbf{i}</math> denotes the imaginary unit <math>\left( \mathbf{i} = \sqrt{-1} \right) \! :</math>

:<math display="block">\begin{bmatrix}

0 & a - \mathbf{i}b & c - \mathbf{i}d \\

a + \mathbf{i}b & 1 & m - \mathbf{i}n \\

c + \mathbf{i}d & m + \mathbf{i}n & 2

\end{bmatrix}.</math>

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, which results in skew-Hermitian matrices.

Here is another useful case of a Hermitian matrix. If a square matrix <math> A </math> equals the product of a matrix <math>B</math> with its conjugate transpose, that is, <math> A = BB^\mathsf{H} ,</math> then <math> A </math> is Hermitian positive semi-definite. Furthermore, if <math> B </math> is non-singular (i.e., row full-rank), then <math> A </math> is positive definite.

Properties

Main diagonal values are real

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

</math> by definition of a Hermitian matrix; so <math>H_{ii} = \overline{H_{ii,</math> and the above follows, as a number can equal its complex conjugate only if their imaginary parts are zero.

Only the main diagonal entries are necessarily real; a Hermitian matrix can have arbitrary complex-valued entries as its off-diagonal elements, as long as its diagonally-opposite entries are complex conjugates.

Symmetric if and only if real

A Hermitian matrix is symmetric if and only if it has only real entries. A symmetric real matrix is simply a special case of a Hermitian matrix.

</math> by definition of a Hermitian matrix. Thus,<br><math>H_{ij} = H_{ji}</math> (i.e., <math>H</math> is symmetric) if and only if <math>\overline{H_{ji = H_{ji}</math> (i.e., <math>H_{ji}</math> is real).

If an anti-symmetric real matrix is multiplied by an imaginary number, then the product is also anti-symmetric but has only imaginary entries. An anti-symmetric imaginary matrix is another special case of a Hermitian matrix.

Normal

Every Hermitian matrix <math>A</math> is a normal matrix; that is to say, <math>AA^\mathsf{H} = A^\mathsf{H}A.</math>

Diagonalizable

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 with order are real, and that has linearly independent eigenvectors. Moreover, a Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues (i.e., zero eigenvalues), it is always possible to find an orthonormal basis of <math>\C^n</math> consisting of eigenvectors of .

Sum of Hermitian matrices is Hermitian

The sum of any two Hermitian matrices is Hermitian.

+ \overline{B_{ji = \overline{A_{ji} + B_{ji = \overline{(A + B)_{ji.</math>

Inverse of Hermitian matrix is Hermitian

The inverse of an invertible Hermitian matrix is Hermitian as well.

Commutative product of Hermitian matrices is Hermitian

The product of two Hermitian matrices and is Hermitian if and only if .

= \overline{B^\mathsf{T} A^\mathsf{T = \overline{B^\mathsf{T \ \overline{A^\mathsf{T = B^\mathsf{H} A^\mathsf{H} = BA.</math><br>Thus, <math>~ (AB)^\mathsf{H} = AB ~</math> if and only if <math>~ AB = BA.</math>

Consequence: If a matrix <math>A</math> is Hermitian, then for any integer <math>m \geq 0, ~ A^m</math> is Hermitian.

is Hermitian

If two matrices <math>A</math> and <math>B</math> are Hermitian, then <math>ABA</math> is also Hermitian.

<sup>H</sup> is real for complex

If a matrix <math>A</math> is Hermitian, then for any complex-valued vector <math>\mathbf{v},</math> the product <math> \mathbf{v}^\mathsf{H} \! A \mathbf{v} </math> is real. Indeed, <math> \mathbf{v}^\mathsf{H} \! A \mathbf{v} = \left( \mathbf{v}^\mathsf{H} \! A \mathbf{v} \right)^\mathsf{H} .</math> 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.

Complex Hermitian matrices form vector space over

The Hermitian complex matrices do not form a vector space over the complex numbers, <math>\C,</math> since the identity matrix <math>I_n</math> is Hermitian, but <math>\mathbf{i} I_n</math> is not. However, the complex Hermitian matrices do form a vector space over the real numbers, <math>\R.</math> In the -dimensional vector space of complex matrices over <math>\R,</math> the complex Hermitian matrices form a subspace of dimension <math>n^2.</math><br> denoting the matrix with a in the position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows:

:<math display="block">E_{jj} \text{ for } 1 \leq j \leq n \quad (n \text{ matrices})</math>

together with the set of matrices of the form

:<math display="block">\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)</math>

and the set of matrices of the form

:<math display="block">\frac{\mathbf{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),</math>

where <math>\mathbf{i}</math> denotes the imaginary unit <math>\Big( \mathbf{i} = \sqrt{-1} \Big).</math>

An example is that the four Pauli matrices form a complete basis for the vector space of all complex Hermitian matrices over <math>\R.</math>

Eigendecomposition

If <math>n</math> orthonormal eigenvectors <math>\mathbf{u}_1, \dots, \mathbf{u}_n</math> of a Hermitian matrix <math>A</math> are chosen and written as the columns of a matrix <math>U,</math> then one eigendecomposition of <math>A</math> is

:<math display="block">A = U \! \varLambda U^\mathsf{H},</math>

where <math>U</math> is a unitary matrixi.e., <math>U U^\mathsf{H} = I = U^\mathsf{H} U,</math> and <math>\varLambda</math> is a diagonal matrix with 's eigenvalues on 's diagonal. Therefore,

:<math display="block">A_{ij} = \mathbf{v}_i \varLambda \mathbf{v}_j \!\! ^\mathsf{H} = \sum_{k=1}^{k=n} \mathbf{v}_{i,k} \lambda_k \overline{\mathbf{v}_{j,k,</math>

where <math>\mathbf{v}_i</math> is the row vector corresponding to the -th row of <math>U, ~</math> <math>\mathbf{v}_{i,k}</math> is the -th coordinate of <math>\mathbf{v}_i,</math> and <math>\lambda_k</math> is the -th eigenvalue on 's diagonal.

Singular values

The singular values of a matrix are the absolute values of its eigenvalues.

Since a Hermitian matrix <math>A</math> has an eigendecomposition <math>A = U \! \varLambda U^\mathsf{H},</math> where <math>U</math> is a unitary matrix (its columns are orthonormal vectorssee above, and its rows are also orthonormal vectors), a singular value decomposition of <math>A</math> is

:<math display="block">A = U |\varLambda| \sgn(\!\varLambda) U^\mathsf{H},</math>

where <math>|\varLambda|</math> and <math>\sgn(\!\varLambda)</math> are diagonal matrices respectively containing the absolute values <math>|\lambda_i|</math> and signs <math>\sgn(\lambda_i)</math> of 's eigenvalues. The matrix <math>\sgn(\!\varLambda) U^\mathsf{H}</math> is unitary, since the rows (not the columns) of <math>U^\mathsf{H}</math> are only getting multiplied by <math>\pm 1.</math> The matrix <math>|\varLambda|</math> contains the singular values of <math>A,</math> namely, the absolute values of 's eigenvalues.

Real determinant

The determinant of a Hermitian matrix is real.

Decomposition into Hermitian and skew-Hermitian matrices

Additional facts related to Hermitian matrices include:

  • The sum of a square matrix and its conjugate transpose <math>\left(A + A^\mathsf{H}\right)</math> is Hermitian.
  • The difference of a square matrix and its conjugate transpose <math>\left(A - A^\mathsf{H}\right)</math> is skew-Hermitian (also called antihermitian). This implies that the commutator of two Hermitian matrices is skew-Hermitian.
  • An arbitrary square matrix can be written as the sum of a Hermitian matrix and a skew-Hermitian matrix . This is known as the Toeplitz decomposition of . <math display="block">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).</math>

Rayleigh quotient

For a complex (or real) matrix <math>M</math> and a non-zero complex (or real) vector <math>\mathbf{v},</math> the Rayleigh quotient, <math>R(M, \mathbf{v}),</math> is defined as the scalar:

:<math display="block">R(M, \mathbf{v}) := \frac{\mathbf{v}^\mathsf{H} \! M \mathbf{v{\mathbf{v}^\mathsf{H} \mathbf{v.</math>

For a real vector <math>\mathbf{v},</math> the conjugate transpose <math>\mathbf{v}^\mathsf{H}</math> reduces to the usual transpose <math>\mathbf{v}^\mathsf{T}.</math>

For any non-zero complex (or real) scalar <math>c,</math>

:<math display="block">R(M, c \mathbf{v}) = R(M, \mathbf{v}).</math>

If <math>M</math> is Hermitian (or symmetric real), then for any non-zero complex vector <math>\mathbf{v}, ~</math> <math>R(M, \mathbf{v})</math> is real.

For a real matrix, the condition of being Hermitian reduces to that of being symmetric.

It can be shown that,