Skew-Hermitian matrix

From formulasearchengine
Jump to navigation Jump to search

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 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:

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 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]

See also

Notes

  1. Template:Harvtxt, §4.1.1; Template:Harvtxt, §3.2
  2. 2.0 2.1 Template:Harvtxt, §4.1.2
  3. Template:Harvtxt, §2.5.2, §2.5.4
  4. Template:Harvtxt, Exercise 3.2.5
  5. 5.0 5.1 Template:Harvtxt, §4.1.1

References

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}.

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}.