Hermitian Matrix

A square matrix is called Hermitian if it is self-adjoint. Therefore, a Hermitian matrix is defined as one for which

(1)

where denotes the conjugate transpose. This is equivalent to the condition

(2)

where denotes the complex conjugate. As a result of this definition, the diagonal elements of a Hermitian matrix are real numbers (since ), while other elements may be complex.

Examples of Hermitian matrices include

(3)

and the Pauli matrices

			(4)
			(5)
			(6)

Examples of Hermitian matrices include

(7)

An integer or real matrix is Hermitian iff it is symmetric. A matrix can be tested to see if it is Hermitian using the Wolfram Language function

  HermitianQ[m_List?MatrixQ] :=
    (m === Conjugate@Transpose@m)

Hermitian matrices have real eigenvalues whose eigenvectors form a unitary basis. For real matrices, Hermitian is the same as symmetric.

Any matrix which is not Hermitian can be expressed as the sum of a Hermitian matrix and a antihermitian matrix using

(8)

Let be a unitary matrix and be a Hermitian matrix. Then the adjoint of a similarity transformation is

			(9)
			(10)
			(11)
			(12)
			(13)

The specific matrix

			(14)
			(15)

where are Pauli matrices, is sometimes called "the" Hermitian matrix.

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