Editing Skew-Hermitian matrix (section)

==Properties==
* The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly zero). Furthermore, skew-Hermitian matrices are [[normal matrix|normal]]. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.<ref>{{harvtxt|Horn|Johnson|1985}}, §2.5.2, §2.5.4</ref>
* All entries on the [[main diagonal]] of a skew-Hermitian matrix have to be pure [[imaginary number|imaginary]]; i.e., on the imaginary axis (the number zero is also considered purely imaginary).<ref>{{harvtxt|Meyer|2000}}, Exercise 3.2.5</ref>
* If <math>A</math> and <math>B</math> are skew-Hermitian, then {{tmath|aA + bB}} is skew-Hermitian for all [[real number|real]] [[scalar (mathematics)|scalars]] <math>a</math> and <math>b</math>.<ref name=HJ85S411>{{harvtxt|Horn|Johnson|1985}}, §4.1.1</ref>
* <math>A</math> is skew-Hermitian ''if and only if'' <math>i A</math> (or equivalently, <math>-i A</math>) is [[Hermitian matrix|Hermitian]].<ref name=HJ85S411/>
*<math>A</math> is skew-Hermitian ''if and only if'' the real part <math>\Re{(A)}</math> is [[skew-symmetric matrix|skew-symmetric]] and the imaginary part <math>\Im{(A)}</math> is [[symmetric matrix|symmetric]].
* If <math>A</math> is skew-Hermitian, then <math>A^k</math> is Hermitian if <math>k</math> is an even integer and skew-Hermitian if <math>k</math> is an odd integer.
* <math>A</math> is skew-Hermitian if and only if <math>\mathbf{x}^\mathsf{H} A \mathbf{y} = -\overline{\mathbf{y}^\mathsf{H} A \mathbf{x}}</math> for all vectors <math>\mathbf x, \mathbf y</math>.
* If <math>A</math> is skew-Hermitian, then the [[matrix exponential]] <math>e^A</math> is [[unitary matrix|unitary]].
* The space of skew-Hermitian matrices forms the [[Lie algebra]] <math>u(n)</math> of the [[Lie group]] <math>U(n)</math>.