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

$A\text{ skew-Hermitian}⟺A^{\mathsf{H}}=-A$

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

$A\text{ skew-Hermitian}⟺a_{ij}=-\overline{a_{ji}}$

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 $$A=\left[\begin{array}{cc}-i& +2+i\\ -2+i& 0\end{array}\right]$$ because $$-A=\left[\begin{array}{cc}i& -2-i\\ 2-i& 0\end{array}\right]=\left[\begin{array}{cc}\overline{-i}& \overline{-2+i}\\ \overline{2+i}& \bar{0}\end{array}\right]={\left[\begin{array}{cc}\overline{-i}& \overline{2+i}\\ \overline{-2+i}& \bar{0}\end{array}\right]}^{\mathsf{T}}=A^{\mathsf{H}}$$

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 $iA$ (or equivalently, $-iA$) is Hermitian.
• $A$ is skew-Hermitian if and only if the real part $\mathrm{\Re }(A)$ is skew-symmetric and the imaginary part $\mathrm{\Im }(A)$ 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 $(A+A^{\mathsf{H}})$ is Hermitian.
• The difference of a square matrix and its conjugate transpose $(A-A^{\mathsf{H}})$ 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 $B$: $$C=A+B\text{with}A=\frac{1}{2}(C+C^{\mathsf{H}})\text{and}B=\frac{1}{2}(C-C^{\mathsf{H}})$$

See also

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

References

• Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6.
• Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8.