Skew-Hermitian matrix

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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 <math>A</math> is skew-Hermitian if it satisfies the relation

where <math>A^\textsf{H}</math> denotes the conjugate transpose of the matrix <math>A</math>. In component form, this means that

</math>

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for all indices <math>i</math> and <math>j</math>, where <math>a_{ij}</math> is the element in the <math>i</math>-th row and <math>j</math>-th column of <math>A</math>, 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 <math>n \times n</math> matrices forms the <math>u(n)</math> Lie algebra, which corresponds to the Lie group U(<var>n</var>). 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 <math>n</math> dimensional complex or real space <math>K^n</math>. If <math>(\cdot\mid\cdot) </math> denotes the scalar product on <math> K^n</math>, then saying <math> A</math> is skew-adjoint means that for all <math>\mathbf u, \mathbf v \in K^n</math> one has <math> (A \mathbf u \mid \mathbf v) = - (\mathbf u \mid A \mathbf v)</math>.

Imaginary numbers can be thought of as skew-adjoint (since they are like <math>1 \times 1</math> matrices), whereas real numbers correspond to self-adjoint operators.

Example

For example, the following matrix is skew-Hermitian

<math display="block"> A = \begin{bmatrix} -i & +2 + i \\ -2 + i & 0 \end{bmatrix}</math>

because

-A =

\begin{bmatrix} i & -2 - i \\ 2 - i & 0 \end{bmatrix} =

\begin{bmatrix}

\overline{-i} & \overline{-2 + i} \\

\overline{2 + i} & \overline{0}

\end{bmatrix} =

\begin{bmatrix}

\overline{-i} & \overline{2 + i} \\

\overline{-2 + i} & \overline{0}

\end{bmatrix}^\mathsf{T} =

A^\mathsf{H}

</math>

Properties

The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly zero). Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.
All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary; i.e., on the imaginary axis (the number zero is also considered purely imaginary).
If <math>A</math> and <math>B</math> are skew-Hermitian, then is skew-Hermitian for all real scalars <math>a</math> and <math>b</math>.
<math>A</math> is skew-Hermitian if and only if <math>i A</math> (or equivalently, <math>-i A</math>) is Hermitian.