Hermitian matrix

 ( Matrices ) 

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

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

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

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

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

Hermitian matrices are named after Charles Hermite, 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 $A^\backslash mathsf\{H\}\; =\; A^\backslash dagger\; =\; A^\backslash ast,$ although in quantum mechanics, $A^\backslash ast$ typically means the complex conjugate only, and not the conjugate transpose.

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

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

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

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Real-valuedness of quadratic forms An $n\backslash times\{\}n$ matrix $A$ is Hermitian if and only if $$\backslash langle\; \backslash mathbf\{v\},\; A\; \backslash mathbf\{v\}\backslash rangle\backslash in\backslash R,\; \backslash quad\; \backslash text\{for\; all\; \}\; \backslash mathbf\{v\}\backslash in\; \backslash mathbb\{C\}^\{n\}.$$

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

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Applications Hermitian matrices are fundamental to quantum mechanics because they describe operators with necessarily real eigenvalues. An eigenvalue $a$ of an operator $\backslash hat\{A\}$ on some quantum state $|\backslash psi\backslash rangle$ 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. 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.

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. 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.

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

See the following example:

$$\backslash begin\{bmatrix\}$$

 0     & a - ib & c-id \\
 a+ib  & 1     & m-in \\
 c+id     &   m+in & 2
     

\end{bmatrix}

The diagonal elements must be real number, 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 number coefficients,

Physics 125 Course Notes at California Institute of Technology 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, $A\; =\; BB^\backslash mathsf\{H\}\; ,$ then $A$ is a Hermitian positive semi-definite matrix. Furthermore, if $B$ is row full-rank, then $A$ is positive definite.

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Properties

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

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

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

So, if a real anti-symmetric matrix is multiplied by a real multiple of the imaginary unit $i,$ then it becomes Hermitian.

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Normal Every Hermitian matrix is a normal matrix. That is to say, $AA^\backslash mathsf\{H\}\; =\; A^\backslash mathsf\{H\}A.$

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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 eigenvectors of a Hermitian matrix with dimension are real, and that has linearly independent . Moreover, a Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an orthogonal basis of consisting of eigenvectors of .

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Sum of Hermitian matrices The sum of any two Hermitian matrices is Hermitian.

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

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Associative product of Hermitian matrices 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. Thus $(AB)^\backslash mathsf\{H\}\; =\; AB$ if and only if $AB\; =\; BA.$
     

Thus is Hermitian if is Hermitian and is an integer.

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ABA Hermitian If A and B are Hermitian, then ABA is also Hermitian.

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is real for complex For an arbitrary complex valued vector the product $\backslash mathbf\{v\}^\backslash mathsf\{H\}\; A\; \backslash mathbf\{v\}$ is real because of $\backslash mathbf\{v\}^\backslash mathsf\{H\}\; A\; \backslash mathbf\{v\}\; =\; \backslash left(\backslash mathbf\{v\}^\backslash mathsf\{H\}\; A\; \backslash mathbf\{v\}\backslash right)^\backslash mathsf\{H\}\; .$ 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.

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Complex Hermitian forms vector space over The Hermitian complex -by- matrices do not form a vector space over the , , since the identity matrix is Hermitian, but is not. However the complex Hermitian matrices do form a vector space over the real numbers . In the -dimensional vector space of complex matrices over , the complex Hermitian matrices form a subspace of dimension . If denotes the -by- matrix with a in the position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows: $$E\_\{jj\}\; \backslash text\{\; for\; \}\; 1\; \backslash leq\; j\; \backslash leq\; n\; \backslash quad\; (n\; \backslash text\{\; matrices\})$$

together with the set of matrices of the form

and the matrices

where $i$ denotes the imaginary unit, $i\; =\; \backslash sqrt\{-1\}~.$

An example is that the four Pauli matrices form a complete basis for the vector space of all complex 2-by-2 Hermitian matrices over .

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Eigendecomposition If orthonormal eigenvectors $\backslash mathbf\{u\}\_1,\; \backslash dots,\; \backslash mathbf\{u\}\_n$ of a Hermitian matrix are chosen and written as the columns of the matrix , then one eigendecomposition of is $A\; =\; U\; \backslash Lambda\; U^\backslash mathsf\{H\}$ where $U\; U^\backslash mathsf\{H\}\; =\; I\; =\; U^\backslash mathsf\{H\}\; U$ and therefore $$A\; =\; \backslash sum\_j\; \backslash lambda\_j\; \backslash mathbf\{u\}\_j\; \backslash mathbf\{u\}\_j\; ^\backslash mathsf\{H\},$$ where $\backslash lambda\_j$ are the eigenvalues on the diagonal of the diagonal matrix $\backslash Lambda.$

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Singular values The singular values of $A$ are the absolute values of its eigenvalues:

Since $A$ has an eigendecomposition $A=U\backslash Lambda\; U^H$, where $U$ is a unitary matrix (its columns are orthonormal vectors; see above), a singular value decomposition of $A$ is $A=U|\backslash Lambda|\backslash text\{sgn\}(\backslash Lambda)U^H$, where $|\backslash Lambda|$ and $\backslash text\{sgn\}(\backslash Lambda)$ are diagonal matrices containing the absolute values $|\backslash lambda|$ and signs $\backslash text\{sgn\}(\backslash lambda)$ of $A$'s eigenvalues, respectively. $\backslash sgn(\backslash Lambda)U^H$ is unitary, since the columns of $U^H$ are only getting multiplied by $\backslash pm\; 1$. $|\backslash Lambda|$ contains the singular values of $A$, namely, the absolute values of its eigenvalues.

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Real determinant 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.)

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Decomposition into Hermitian and skew-Hermitian matrices Additional facts related to Hermitian matrices include:

• The sum of a square matrix and its conjugate transpose $\backslash left(A\; +\; A^\backslash mathsf\{H\}\backslash right)$ is Hermitian.
• The difference of a square matrix and its conjugate transpose $\backslash left(A\; -\; A^\backslash mathsf\{H\}\backslash right)$ 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 .
  (2025). 9780521839402, Cambridge University Press. ISBN 9780521839402
  $$C\; =\; A\; +\; B\; \backslash quad\backslash text\{with\}\backslash quad\; A\; =\; \backslash frac\{1\}\{2\}\backslash left(C\; +\; C^\backslash mathsf\{H\}\backslash right)\; \backslash quad\backslash text\{and\}\backslash quad\; B\; =\; \backslash frac\{1\}\{2\}\backslash left(C\; -\; C^\backslash mathsf\{H\}\backslash right)$$

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Rayleigh quotient In mathematics, for a given complex Hermitian matrix and nonzero vector , the Rayleigh quotientAlso known as the Rayleigh–Ritz ratio; named after Walther Ritz and Lord Rayleigh. $R(M,\; \backslash mathbf\{x\}),$ is defined as:Parlet B. N. The symmetric eigenvalue problem, SIAM, Classics in Applied Mathematics,1998 $$R(M,\; \backslash mathbf\{x\})\; :=\; \backslash frac\{\backslash mathbf\{x\}^\backslash mathsf\{H\}\; M\; \backslash mathbf\{x\}\}\{\backslash mathbf\{x\}^\backslash mathsf\{H\}\; \backslash mathbf\{x\}\}.$$

For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose $\backslash mathbf\{x\}^\backslash mathsf\{H\}$ to the usual transpose $\backslash mathbf\{x\}^\backslash mathsf\{T\}.$ $R(M,\; c\; \backslash mathbf\; x)\; =\; R(M,\; \backslash mathbf\; x)$ 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 $\backslash lambda\_\backslash min$ (the smallest eigenvalue of M) when $\backslash mathbf\; x$ is $\backslash mathbf\; v\_\backslash min$ (the corresponding eigenvector). Similarly, $R(M,\; \backslash mathbf\; x)\; \backslash leq\; \backslash lambda\_\backslash max$ and $R(M,\; \backslash mathbf\; v\_\backslash max)\; =\; \backslash lambda\_\backslash max\; .$

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, $\backslash lambda\_\backslash max$ is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to associates the Rayleigh quotient for a fixed and varying through the algebra would be referred to as "vector state" of the algebra.

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See also

• (anti-Hermitian matrix)

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External links

• Visualizing Hermitian Matrix as An Ellipse with Dr. Geo , by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.

Categories: Matrices

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