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

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.


For example, the following matrix is skew-Hermitian:


See also


  1. Horn & Johnson (1985), §4.1.1; Meyer (2000), §3.2
  2. 1 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. 1 2 Horn & Johnson (1985), §4.1.1


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