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 -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and :
or in matrix form:
Hermitian matrices can be understood as the complex extension of real symmetric matrices.
If the conjugate transpose of a matrix formula_2 is denoted by formula_3, 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 formula_4, although note that in quantum mechanics, formula_5 typically means the complex conjugate only, and not the conjugate transpose.
Alternative characterizations.
Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:
Equality with the adjoint.
A square matrix formula_2 is Hermitian if and only if it is equal to its adjoint, that is, it satisfies
formula_7
for any pair of vectors formula_8, where formula_9 denotes the inner product operation.
This is also the way that the more general concept of self-adjoint operator is defined.
Reality of quadratic forms.
A square matrix formula_2 is Hermitian if and only if
formula_11
Spectral properties.
A square matrix formula_2 is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues.
Applications.
Hermitian matrices are fundamental to the quantum theory of matrix mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. 
Examples.
In this section, the conjugate transpose of matrix formula_13 is denoted as formula_14, the transpose of matrix formula_13 is denoted as formula_16 and conjugate of matrix formula_13 is denoted as formula_18.
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, which results in skew-Hermitian matrices.
Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix formula_13 equals the multiplication of a matrix and its conjugate transpose, that is, formula_21, then formula_13 is a Hermitian positive semi-definite matrix. Furthermore, if formula_23 is row full-rank, then formula_13 is positive definite.
Decomposition into Hermitian and skew-Hermitian.
Additional facts related to Hermitian matrices include:
Rayleigh quotient.
In mathematics, for a given complex Hermitian matrix "M" and nonzero vector x, the Rayleigh quotient formula_59, is defined as:
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose formula_61 to the usual transpose formula_62. Note that formula_63 for any non-zero real scalar formula_64. Also, 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 formula_65 (the smallest eigenvalue of M) when formula_66 is formula_67 (the corresponding eigenvector). Similarly, formula_68 and formula_69.
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, formula_70 is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to associates the Rayleigh quotient for a fixed and varying through the algebra would be referred to as "vector state" of the algebra.