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.[1] That is, the matrix [math]\displaystyle{ A }[/math] is skew-Hermitian if it satisfies the relation
[math]\displaystyle{ A \text{ skew-Hermitian} \quad \iff \quad A^\mathsf{H} = -A }[/math]
where [math]\displaystyle{ A^\textsf{H} }[/math] denotes the conjugate transpose of the matrix [math]\displaystyle{ A }[/math]. In component form, this means that
[math]\displaystyle{ A \text{ skew-Hermitian} \quad \iff \quad a_{ij} = -\overline{a_{ji}} }[/math]
for all indices [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math], where [math]\displaystyle{ a_{ij} }[/math] is the element in the [math]\displaystyle{ i }[/math]-th row and [math]\displaystyle{ j }[/math]-th column of [math]\displaystyle{ A }[/math], 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.[2] The set of all skew-Hermitian [math]\displaystyle{ n \times n }[/math] matrices forms the [math]\displaystyle{ u(n) }[/math] 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 [math]\displaystyle{ n }[/math] dimensional complex or real space [math]\displaystyle{ K^n }[/math]. If [math]\displaystyle{ (\cdot\mid\cdot) }[/math] denotes the scalar product on [math]\displaystyle{ K^n }[/math], then saying [math]\displaystyle{ A }[/math] is skew-adjoint means that for all [math]\displaystyle{ \mathbf u, \mathbf v \in K^n }[/math] one has [math]\displaystyle{ (A \mathbf u \mid \mathbf v) = - (\mathbf u \mid A \mathbf v) }[/math].
Imaginary numbers can be thought of as skew-adjoint (since they are like [math]\displaystyle{ 1 \times 1 }[/math] matrices), whereas real numbers correspond to self-adjoint operators.
Example
For example, the following matrix is skew-Hermitian [math]\displaystyle{ A = \begin{bmatrix} -i & +2 + i \\ -2 + i & 0 \end{bmatrix} }[/math] because [math]\displaystyle{ -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} }[/math]
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.[3]
- 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).[4]
- If [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] are skew-Hermitian, then [math]\displaystyle{ aA + bB }[/math] is skew-Hermitian for all real scalars [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math].[5]
- [math]\displaystyle{ A }[/math] is skew-Hermitian if and only if [math]\displaystyle{ i A }[/math] (or equivalently, [math]\displaystyle{ -i A }[/math]) is Hermitian.[5]
- [math]\displaystyle{ A }[/math] is skew-Hermitian if and only if the real part [math]\displaystyle{ \Re{(A)} }[/math] is skew-symmetric and the imaginary part [math]\displaystyle{ \Im{(A)} }[/math] is symmetric.
- If [math]\displaystyle{ A }[/math] is skew-Hermitian, then [math]\displaystyle{ A^k }[/math] is Hermitian if [math]\displaystyle{ k }[/math] is an even integer and skew-Hermitian if [math]\displaystyle{ k }[/math] is an odd integer.
- [math]\displaystyle{ A }[/math] is skew-Hermitian if and only if [math]\displaystyle{ \mathbf{x}^\mathsf{H} A \mathbf{y} = -\mathbf{y}^\mathsf{H} A \mathbf{x} }[/math] for all vectors [math]\displaystyle{ \mathbf x, \mathbf y }[/math].
- If [math]\displaystyle{ A }[/math] is skew-Hermitian, then the matrix exponential [math]\displaystyle{ e^A }[/math] is unitary.
- The space of skew-Hermitian matrices forms the Lie algebra [math]\displaystyle{ u(n) }[/math] of the Lie group [math]\displaystyle{ U(n) }[/math].
Decomposition into Hermitian and skew-Hermitian
- The sum of a square matrix and its conjugate transpose [math]\displaystyle{ \left(A + A^\mathsf{H}\right) }[/math] is Hermitian.
- The difference of a square matrix and its conjugate transpose [math]\displaystyle{ \left(A - A^\mathsf{H}\right) }[/math] is skew-Hermitian. This implies that the commutator of two Hermitian matrices is skew-Hermitian.
- An arbitrary square matrix [math]\displaystyle{ C }[/math] can be written as the sum of a Hermitian matrix [math]\displaystyle{ A }[/math] and a skew-Hermitian matrix [math]\displaystyle{ B }[/math]: [math]\displaystyle{ 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) }[/math]
See also
Notes
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, http://www.matrixanalysis.com/.
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