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 $$A$$ is skew-Hermitian if it satisfies the relation

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

for all indices $$i$$ and $$j$$, where $$a_{ij}$$ is the element in the $$i$$-th row and $$j$$-th column 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. The set of all skew-Hermitian $$n \times n$$ matrices forms 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.

Note that the adjoint of an operator depends on the scalar product considered on the $$n$$ dimensional complex or real space $$K^n$$. If $$(\cdot\mid\cdot) $$ denotes the scalar product on $$ K^n$$, then saying $$ A$$ is skew-adjoint means that for all $$\mathbf u, \mathbf v \in K^n$$ one has $$ (A \mathbf u \mid \mathbf v) = - (\mathbf u \mid A \mathbf v)$$.

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

Example
For example, the following matrix is skew-Hermitian $$ A = \begin{bmatrix} -i & +2 + i \\ -2 + i & 0 \end{bmatrix}$$ because $$ -A = \begin{bmatrix} i & -2 - i \\ 2 - i & 0 \end{bmatrix} = \begin{bmatrix} \overline{-i}   & \overline{-2 + i} \\ \overline{2 + i} & \overline{0} \end{bmatrix} = \begin{bmatrix} \overline{-i}    & \overline{2 + i} \\ \overline{-2 + i} &    \overline{0} \end{bmatrix}^\mathsf{T} = A^\mathsf{H} $$

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 $$A$$ and $$B$$ are skew-Hermitian, then $aA + bB$ is skew-Hermitian for all real scalars $$a$$ and $$b$$.
 * $$A$$ is skew-Hermitian if and only if $$i A$$ (or equivalently, $$-i A$$) is Hermitian.
 * $$A$$ is skew-Hermitian if and only if the real part $$\Re{(A)}$$ is skew-symmetric and the imaginary part $$\Im{(A)}$$ is symmetric.
 * If $$A$$ is skew-Hermitian, then $$A^k$$ is Hermitian if $$k$$ is an even integer and skew-Hermitian if $$k$$ is an odd integer.
 * $$A$$ is skew-Hermitian if and only if $$\mathbf{x}^\mathsf{H} A \mathbf{y} = -\overline{\mathbf{y}^\mathsf{H} A \mathbf{x}}$$ for all vectors $$\mathbf x, \mathbf y$$.
 * If $$A$$ is skew-Hermitian, then the matrix exponential $$e^A$$ is unitary.
 * The space of skew-Hermitian matrices forms the Lie algebra $$u(n)$$ of the Lie group $$U(n)$$.

Decomposition into Hermitian and skew-Hermitian

 * The sum of a square matrix and its conjugate transpose $$\left(A + A^\mathsf{H}\right)$$ is Hermitian.
 * The difference of a square matrix and its conjugate transpose $$\left(A - A^\mathsf{H}\right)$$ is skew-Hermitian. This implies that the commutator of two Hermitian matrices is skew-Hermitian.
 * An arbitrary square matrix $$C$$ can be written as the sum of a Hermitian matrix $$A$$ and a skew-Hermitian matrix $$B$$: $$C = A + B \quad\mbox{with}\quad A = \frac{1}{2}\left(C + C^\mathsf{H}\right) \quad\mbox{and}\quad B = \frac{1}{2}\left(C - C^\mathsf{H}\right)$$