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

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

for all indices $i$ and $j$, where $a_$ 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., §4.1.2 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
$A = \begin -i & +2 + i \\ -2 + i & 0 \end$
because
$-A = \begin i & -2 - i \\ 2 - i & 0 \end = \begin \overline & \overline \\ \overline & \overline \end = \begin \overline & \overline \\ \overline & \overline \end^\mathsf = A^\mathsf$

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)., Exercise 3.2.5
* If $A$ and $B$ are skew-Hermitian, then is skew-Hermitian for all real scalars $a$ and $b$., §4.1.1
* $A$ 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 $\Re$ is skew-symmetric and the imaginary part $\Im$ 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 $\mathbf^\mathsf A \mathbf = -\overline$ 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 $\left(A + A^\mathsf\right)$ is Hermitian.
* The difference of a square matrix and its conjugate transpose $\left(A - A^\mathsf\right)$ 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 $B$: $C = A + B \quad\mbox\quad A = \frac\left(C + C^\mathsf\right) \quad\mbox\quad B = \frac\left(C - C^\mathsf\right)$

See also

* Bivector (complex)
* Hermitian matrix
* Normal matrix
* Skew-symmetric matrix
* Unitary matrix

Notes

References

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{{Matrix classes

Matrices (mathematics)
Abstract algebra
Linear algebra