skew-Hermitian matrix

{{Short description|Matrix whose conjugate transpose is its negative (additive inverse)}}

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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.{{harvtxt|Horn|Johnson|1985}}, §4.1.1; {{harvtxt|Meyer|2000}}, §3.2 That is, the matrix $A$ is skew-Hermitian if it satisfies the relation

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

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|equation = $A \text{ skew-Hermitian} \quad \iff \quad a_{ij} = -\overline{a_{ji}}$

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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.{{harvtxt|Horn|Johnson|1985}}, §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

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

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.{{harvtxt|Horn|Johnson|1985}}, §2.5.2, §2.5.4
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).{{harvtxt|Meyer|2000}}, Exercise 3.2.5
If $A$ and $B$ are skew-Hermitian, then {{tmath|aA + bB}} is skew-Hermitian for all real scalars $a$ and $b$ .{{harvtxt|Horn|Johnson|1985}}, §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 $\Re{(A)}$ 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 $B$ : $C = A + B \quad\mbox{with}\quad A = \frac{1}{2}\left(C + C^\mathsf{H}\right) \quad\mbox{and}\quad B = \frac{1}{2}\left(C - C^\mathsf{H}\right)$

Notes

References

{{Citation | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher=Cambridge University Press | isbn=978-0-521-38632-6 | year=1985}}.
{{Citation | last1=Meyer | first1=Carl D. | title=Matrix Analysis and Applied Linear Algebra | url=http://www.matrixanalysis.com/ | publisher=SIAM | isbn=978-0-89871-454-8 | year=2000}}.

Category:Matrices (mathematics)

Category:Abstract algebra

Category:Linear algebra