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

$A^\dagger = -A,\;$

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

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

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:

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

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 $aA + bB$ 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 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]

$C = A+B \quad\mbox{with}\quad A = \frac{1}{2}(C + C^\dagger) \quad\mbox{and}\quad B = \frac{1}{2}(C - C^\dagger).$

• 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

• Bivector (complex)
• Hermitian matrix
• Normal matrix
• Skew-symmetric matrix
• Unitary matrix

Notes

1. Horn & Johnson (1985), §4.1.1; Meyer (2000), §3.2
2. Horn & Johnson (1985), §4.1.2
3. Horn & Johnson (1985), §2.5.2, §2.5.4
4. Meyer (2000), Exercise 3.2.5
5. 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.