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

Hermitian matrix. A square matrix such that aij is the complex conjugate of aji for all elements aij 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 aij 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).