Skew-Hermitian matrix


In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix.[1] That is, the matrix is skew-Hermitian if it satisfies the relation

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

for all indices and , where is the element in the -th row and -th column of , 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] The set of all skew-Hermitian matrices forms the 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.

Note that the adjoint of an operator depends on the scalar product considered on the dimensional complex or real space . If denotes the scalar product on , then saying is skew-adjoint means that for all one has .

Imaginary numbers can be thought of as skew-adjoint (since they are like matrices), whereas real numbers correspond to self-adjoint operators.

Example edit

For example, the following matrix is skew-Hermitian


Properties edit

  • The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly 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   and   are skew-Hermitian, then   is skew-Hermitian for all real scalars   and  .[5]
  •   is skew-Hermitian if and only if   (or equivalently,  ) is Hermitian.[5]
  •   is skew-Hermitian if and only if the real part   is skew-symmetric and the imaginary part   is symmetric.
  • If   is skew-Hermitian, then   is Hermitian if   is an even integer and skew-Hermitian if   is an odd integer.
  •   is skew-Hermitian if and only if   for all vectors  .
  • If   is skew-Hermitian, then the matrix exponential   is unitary.
  • The space of skew-Hermitian matrices forms the Lie algebra   of the Lie group  .

Decomposition into Hermitian and skew-Hermitian edit

  • The sum of a square matrix and its conjugate transpose   is Hermitian.
  • The difference of a square matrix and its conjugate transpose   is skew-Hermitian. This implies that the commutator of two Hermitian matrices is skew-Hermitian.
  • An arbitrary square matrix   can be written as the sum of a Hermitian matrix   and a skew-Hermitian matrix  :

See also edit

Notes edit

  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. ^ a b Horn & Johnson (1985), §4.1.1

References edit

  • 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.