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. Other, equivalent notations in common use are although in quantum mechanics, typically means the complex conjugate only, and not the conjugate transpose. Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below: A square matrix is Hermitian if and only if it is equal to its adjoint, 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. An matrix is Hermitian if and only if A square matrix is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues.' 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 necessitates the need for operators with 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 Hermitian matrices include the Pauli matrices, the Gell-Mann matrices and their generalizations.