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. That is, the matrix formula_1 is skew-Hermitian if it satisfies the relation
where formula_2 denotes the conjugate transpose of the matrix formula_1. In component form, this means that
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for all indices formula_4 and formula_5, where formula_6 is the element in the formula_5-th row and formula_4-th column of formula_1, 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 formula_10 matrices forms the formula_11 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 formula_12 dimensional complex or real space formula_13. If formula_14 denotes the scalar product on formula_15, then saying formula_16 is skew-adjoint means that for all formula_17 one has formula_18.
Imaginary numbers can be thought of as skew-adjoint (since they are like formula_19 matrices), whereas real numbers correspond to self-adjoint operators.
Example.
For example, the following matrix is skew-Hermitian
because