Samiksha Jaiswal (Editor)

Skew Hermitian matrix

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In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or antihermitian if its conjugate transpose is equal to its negative. That is, the matrix A is skew-Hermitian if it satisfies the relation

Contents

A † = − A ,

where † denotes the conjugate transpose of a matrix. In component form, this means that

a i , j = − a j , i ¯ ,

for all i and j, where ai,j is the i,j-th entry of A, and the overline denotes complex conjugation.

Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers. All skew-Hermitian n×n matrices form the u(n) Lie algebra, which corresponds to the Lie group U(n). The concept can be generalized to include linear transformations of any complex vector space with a sesquilinear norm.

Example

For example, the following matrix is skew-Hermitian:

[ − i 2 + i − ( 2 − i ) 0 ]

Properties

  • The eigenvalues of a skew-Hermitian matrix are all purely imaginary or zero. Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.
  • All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, i.e., on the imaginary axis (the number zero is also considered purely imaginary).
  • If A, B are skew-Hermitian, then a A + b B is skew-Hermitian for all real scalars a and b.
  • If A is skew-Hermitian, then both i A and −i A are Hermitian.
  • If A is skew-Hermitian, then Ak is Hermitian if k is an even integer and skew-Hermitian if k is an odd integer.
  • An arbitrary (square) matrix C can uniquely be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B:
  • If A is skew-Hermitian, then eA is unitary.
  • The space of skew-Hermitian matrices forms the Lie algebra u(n) of the Lie group U(n).

    • References

    Skew-Hermitian matrix Wikipedia
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