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

$A skew-Hermitian ⟺ A^{𝖧} = - A$

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

$A skew-Hermitian ⟺ a_{i j} = - \overline{a_{j i}}$

for all indices $i$ and $j$ , where $a_{i j}$ 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 Hermitian product.

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

In dimension 1, the skew-adjoint operators are exactly the imaginary numbers, whereas real numbers correspond to self-adjoint operators.

Example

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

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 ⁠ $a A + b B$ ⁠ 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 $ℜ (A)$ is skew-symmetric and the imaginary part $ℑ (A)$ 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 $𝐱^{𝖧} A 𝐲 = - \overline{𝐲^{𝖧} A 𝐱}$ for all vectors $𝐱, 𝐲$ .
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 $(A + A^{𝖧})$ is Hermitian.
The difference of a square matrix and its conjugate transpose $(A - A^{𝖧})$ 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 $B$ : $C = A + B with A = \frac{1}{2} (C + C^{𝖧}) and B = \frac{1}{2} (C - C^{𝖧})$

Notes

↑ Horn & Johnson (1985), §4.1.1; Meyer (2000), §3.2
↑ Horn & Johnson (1985), §4.1.2
↑ Horn & Johnson (1985), §2.5.2, §2.5.4
↑ Meyer (2000), Exercise 3.2.5
1 2 Horn & Johnson (1985), §4.1.1

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.

[1] Horn & Johnson (1985), §4.1.1; Meyer (2000), §3.2

[HJ85S412-2] Horn & Johnson (1985), §4.1.2

[3] Horn & Johnson (1985), §2.5.2, §2.5.4

[4] Meyer (2000), Exercise 3.2.5

[HJ85S411-5] 1 2 Horn & Johnson (1985), §4.1.1

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