||It has been suggested that Skew-Hermitian be merged into this article. (Discuss) Proposed since June 2016.|
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. 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 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.
For example, the following matrix is skew-Hermitian:
- 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 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).
- Horn & Johnson (1985), §4.1.1; Meyer (2000), §3.2
- Horn & Johnson (1985), §4.1.2
- Horn & Johnson (1985), §2.5.2, §2.5.4
- Meyer (2000), Exercise 3.2.5
- Horn & Johnson (1985), §4.1.1