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.


For example, the following matrix is skew-Hermitian because


Decomposition into Hermitian and skew-Hermitian

See also


  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