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:

or in matrix form:

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,[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 although in quantum mechanics, typically means the complex conjugate only, and not the conjugate transpose.

Alternative characterizations edit

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

Equality with the adjoint edit

A square matrix   is Hermitian if and only if it is equal to its conjugate transpose, that is, it satisfies

for any pair of vectors   where   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 edit

An   matrix   is Hermitian if and only if


Spectral properties edit

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

Applications edit

Hermitian matrices are fundamental to quantum mechanics because they describe operators with necessarily real eigenvalues. An eigenvalue   of an operator   on some quantum state   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   is denoted as   the transpose of matrix   is denoted as   and conjugate of matrix   is denoted as  

See the following example:


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   equals the product of a matrix with its conjugate transpose, that is,   then   is a Hermitian positive semi-definite matrix. Furthermore, if   is row full-rank, then   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.


By definition of the Hermitian matrix

so for i = j the above follows.

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.

Symmetric edit

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


  by definition. Thus   (matrix symmetry) if and only if   (  is real).

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

Normal edit

Every Hermitian matrix is a normal matrix. That is to say,  



Diagonalizable edit

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 A with dimension n are real, and that A has n linearly independent eigenvectors. 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 Cn consisting of n eigenvectors of A.

Sum of Hermitian matrices edit

The sum of any two Hermitian matrices is Hermitian.


as claimed.

Inverse is Hermitian edit

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


If   then   so   as claimed.

Associative product of Hermitian matrices edit

The product of two Hermitian matrices A and B is Hermitian if and only if AB = BA.


Thus   if and only if  

Thus An is Hermitian if A is Hermitian and n is an integer.

ABA Hermitian edit

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



vHAv is real for complex v edit

For an arbitrary complex valued vector v 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.

Complex Hermitian forms vector space over R edit

The Hermitian complex n-by-n matrices do not form a vector space over the complex numbers, C, since the identity matrix In is Hermitian, but iIn is not. However the complex Hermitian matrices do form a vector space over the real numbers R. In the 2n2-dimensional vector space of complex n × n matrices over R, the complex Hermitian matrices form a subspace of dimension n2. If Ejk denotes the n-by-n matrix with a 1 in the j,k position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows:


together with the set of matrices of the form


and the matrices


where   denotes the imaginary unit,  

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 R.

Eigendecomposition edit

If n orthonormal eigenvectors   of a Hermitian matrix are chosen and written as the columns of the matrix U, then one eigendecomposition of A is   where   and therefore

where   are the eigenvalues on the diagonal of the diagonal matrix  

Singular values edit

The singular values of   are the absolute values of its eigenvalues:

Since   has an eigendecomposition  , where   is a unitary matrix (its columns are orthonormal vectors; see above), a singular value decomposition of   is  , where   and   are diagonal matrices containing the absolute values   and signs   of  's eigenvalues, respectively.   is unitary, since the columns of   are only getting multiplied by  .   contains the singular values of  , namely, the absolute values of its eigenvalues.[8]

Real determinant edit

The determinant of a Hermitian matrix is real:


  Therefore if  

(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 matrices edit

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 the commutator 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.[9]: 227 

Rayleigh quotient edit

In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient[10]   is defined as:[9]: p. 234 [11]


For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose   to the usual transpose     for any non-zero real scalar   Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.

It can be shown[9] that, for a given matrix, the Rayleigh quotient reaches its minimum value   (the smallest eigenvalue of M) when   is   (the corresponding eigenvector). Similarly,   and  

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,   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.

See also edit

References edit

  1. ^ Archibald, Tom (2010-12-31), Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.), "VI.47 Charles Hermite", The Princeton Companion to Mathematics, Princeton University Press, pp. 773–773, doi:10.1515/9781400830398.773a, ISBN 978-1-4008-3039-8, retrieved 2023-11-15
  2. ^ Ribeiro, Alejandro. "Signal and Information Processing" (PDF).
  4. ^ Lau, Ivan. "Hermitian Spectral Theory of Mixed Graphs" (PDF).
  5. ^ Liu, Jianxi; Li, Xueliang (February 2015). "Hermitian-adjacency matrices and Hermitian energies of mixed graphs". Linear Algebra and its Applications. 466: 182–207. doi:10.1016/j.laa.2014.10.028.
  6. ^ Frankel, Theodore (2004). The Geometry of Physics: an introduction. Cambridge University Press. p. 652. ISBN 0-521-53927-7.
  7. ^ Physics 125 Course Notes at California Institute of Technology
  8. ^ Trefethan, Lloyd N.; Bau, III, David (1997). Numerical linear algebra. Philadelphia, PA, USA: SIAM. p. 34. ISBN 0-89871-361-7.
  9. ^ a b c Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402.
  10. ^ Also known as the Rayleigh–Ritz ratio; named after Walther Ritz and Lord Rayleigh.
  11. ^ Parlet B. N. The symmetric eigenvalue problem, SIAM, Classics in Applied Mathematics,1998

External links edit

  • "Hermitian matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Visualizing Hermitian Matrix as An Ellipse with Dr. Geo, by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.
  • "Hermitian Matrices". MathPages.com.