Website owner: James Miller
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).
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