# Skew-Hermitian 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 its negative.^{[1]} That is, the matrix *A* is skew-Hermitian if it satisfies the relation

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

for all *i* and *j*, where *a*_{i,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:

## 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 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*A*^{k}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]}

- If
*A*is skew-Hermitian, then e^{A}is unitary. - The space of skew-Hermitian matrices forms the Lie algebra u(
*n*) of the Lie group U(*n*).

## See also

## Notes

- ↑ Horn & Johnson (1985), §4.1.1; Meyer (2000), §3.2
- 1 2 Horn & Johnson (1985), §4.1.2
- ↑ Horn & Johnson (1985), §2.5.2, §2.5.4
- ↑ Meyer (2000), Exercise 3.2.5
- 1 2 Horn & Johnson (1985), §4.1.1

## References

- Horn, Roger A.; Johnson, Charles R. (1985),
*Matrix Analysis*, Cambridge University Press, ISBN 978-0-521-38632-6. - Meyer, Carl D. (2000),
*Matrix Analysis and Applied Linear Algebra*, SIAM, ISBN 978-0-89871-454-8.