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, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues.


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 Pauli matrices, Gell-Mann matrices and their generalizations are Hermitian. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients,[1][2] which results in skew-Hermitian matrices (see below).

Here we offer another useful Hermitian matrix using an abstract example. If a square matrix equals the multiplication of a matrix and its conjugate transpose, that is, , then is a Hermitian positive semi-definite matrix. Furthermore, if is row full-rank, then is positive definite.


for (n matrices)
together with the set of matrices of the form
for (n2n/2 matrices)
and the matrices
for (n2n/2 matrices)
where denotes the complex number , known as the imaginary unit.
where are the eigenvalues on the diagonal of the diagonal matrix .

Further properties

Additional facts related to Hermitian matrices include:

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.)

Rayleigh quotient

Main article: Rayleigh quotient

In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient[3] , is defined as:[4][5]

For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose to the usual transpose . Note that for any non-zero real scalar . 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 (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-Ritz 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


  1. Frankel, Theodore (2004). The geometry of physics: an introduction. Cambridge University Press. p. 652. ISBN 0-521-53927-7.
  2. Physics 125 Course Notes at California Institute of Technology
  3. Also known as the Rayleigh–Ritz ratio; named after Walther Ritz and Lord Rayleigh.
  4. Horn, R. A. and C. A. Johnson. 1985. Matrix Analysis. Cambridge University Press. pp. 176180.
  5. Parlet B. N. The symmetric eigenvalue problem, SIAM, Classics in Applied Mathematics,1998

External links

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