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 the negative of the original matrix. 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. The set of all skew-Hermitian matrices forms the Lie algebra, which corresponds to the Lie group U. 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, whereas real numbers correspond to self-adjoint operators.

Example

For example, the following matrix is skew-Hermitian
because

Properties