# Skew-Hermitian matrix

(Redirected from Antihermitian)
For matrices with antisymmetry over the real number field, see Skew-symmetric matrix.

In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or antihermitian if its conjugate transpose is equal to the original matrix, with all the entries being of opposite sign.[1] That is, the matrix A is skew-Hermitian if it satisfies the relation

${\displaystyle A^{\dagger }=-A,\;}$

where ${\displaystyle \dagger }$ denotes the conjugate transpose of a matrix. In component form, this means that

${\displaystyle a_{i,j}=-{\overline {a_{j,i}}},}$

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.[2] 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:

${\displaystyle {\begin{bmatrix}-i&2+i\\-(2-i)&0\end{bmatrix}}}$

## 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.[3]
• 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).[4]
• If A, B are skew-Hermitian, then ${\displaystyle aA+bB}$ is skew-Hermitian for all real scalars a and b.[5]
• If A is skew-Hermitian, then both i A and −i A are Hermitian.[5]
• 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:[2]
${\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}(C+C^{\dagger })\quad {\mbox{and}}\quad B={\frac {1}{2}}(C-C^{\dagger }).}$