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 Hermitian matrix, Skew-Hermitian matrix, Hermitian conjugate of a matrix

Hermitian matrix. A square matrix such that is the complex conjugate of for all elements of the matrix i.e.a matrix in which corresponding elements with respect to the diagonal are conjugates of each other. The diagonal elements are always real numbers.

Example.

 A Hermitian matrix can also be defined as a square matrix A in which the transpose of the conjugate of A is equal to A i.e. where

Both definitions are equivalent.

Skew-Hermitian matrix. A square matrix such that

for all elements of the matrix. The diagonal elements are either zeros or pure imaginaries.

 Example.                    

A Skew-Hermitian matrix can also be defined as a square matrix A in which

                       .

Both definitions are equivalent.

Hermitian conjugate of a matrix. The transpose of the conjugate of a matrix. For a square matrix A it is the matrix

                             .

Theorems.

Theorem 1. If A is a square matrix then

is Hermitian and

 is skew-Hermitian.

Theorem 2. Every square matrix with complex elements can be written as the sum A = B + C of a Hermitian matrix

and a skew-Hermitian matrix

References.

  Ayres. Matrices (Schaum).