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:

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

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

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

Hermitian matrices are fundamental to the quantum theory of matrix mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.

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,[1][2] which results in skew-Hermitian matrices.

(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)

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.