Hermitian matrix

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In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose—that is, the element in the Template:Mvar-th row and Template:Mvar-th column is equal to the complex conjugate of the element in the Template:Mvar-th row and Template:Mvar-th column, for all indices Template:Mvar and Template:Mvar:

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 having eigenvalues always real.


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.


  • The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are necessarily real, because they have to be equal to their complex conjugate. A matrix that has only real entries is Hermitian if and only if it is a symmetric matrix, i.e., if it is symmetric with respect to the main diagonal. A real and symmetric matrix is simply a special case of a Hermitian matrix.
  • The Hermitian complex Template:Mvar-by-Template:Mvar matrices do not form a vector space over the complex numbers, 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 Template:Mvar-by-Template:Mvar matrix with a 1 in the j,k position and zeros elsewhere, a basis can be described as follows:
for (Template:Mvar matrices)
together with the set of matrices of the form
for ({{ safesubst:#invoke:Unsubst||$B=n2n/2}} matrices)
and the matrices
for ({{ safesubst:#invoke:Unsubst||$B=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

{{safesubst:#invoke:anchor|main}}Additional facts related to Hermitian matrices include:

  • 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.)

Rayleigh quotient

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See also


  1. {{#invoke:citation/CS1|citation |CitationClass=book }}
  2. Physics 125 Course Notes at California Institute of Technology

External links

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