Editing 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 number|complex]] entries is said to be '''skew-Hermitian''' or '''anti-Hermitian''' if its [[conjugate transpose]] is the negative of the original matrix.<ref>{{harvtxt|Horn|Johnson|1985}}, §4.1.1; {{harvtxt|Meyer|2000}}, §3.2</ref> That is, the matrix <math>A</math> is skew-Hermitian if it satisfies the relation

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|equation = <math>A \text{ skew-Hermitian} \quad \iff \quad A^\mathsf{H} = -A</math>
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where <math>A^\textsf{H}</math> denotes the conjugate transpose of the matrix <math>A</math>.  In component form, this means that

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|equation = <math>A \text{ skew-Hermitian} \quad \iff \quad a_{ij} = -\overline{a_{ji}}</math>
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for all indices <math>i</math> and <math>j</math>, where <math>a_{ij}</math> is the element in the <math>i</math>-th row and <math>j</math>-th column of <math>A</math>, and the overline denotes [[complex conjugate|complex conjugation]].

Skew-Hermitian matrices can be understood as the complex versions of real [[Skew-symmetric matrix|skew-symmetric matrices]], or as the matrix analogue of the purely imaginary numbers.<ref name=HJ85S412>{{harvtxt|Horn|Johnson|1985}}, §4.1.2</ref>  The set of all skew-Hermitian <math>n \times n</math> matrices forms the <math>u(n)</math> [[Lie algebra]], which corresponds to the Lie group [[Unitary group|U(<var>n</var>)]]. The concept can be generalized to include [[linear transformation]]s of any [[complex number|complex]] [[vector space]] with a [[sesquilinear]] [[Norm (mathematics)|norm]].

Note that the [[adjoint operator|adjoint]] of an operator depends on the [[scalar product]] considered on the <math>n</math> dimensional complex or real space <math>K^n</math>. If <math>(\cdot\mid\cdot) </math> denotes the scalar product on <math> K^n</math>, then saying <math> A</math> is skew-adjoint means that for all <math>\mathbf u, \mathbf v \in K^n</math> one has <math> (A \mathbf u \mid \mathbf v) = - (\mathbf u \mid A \mathbf v)</math>.

[[Imaginary number]]s can be thought of as skew-adjoint (since they are like <math>1 \times 1</math> matrices), whereas [[real number]]s correspond to [[self-adjoint]] operators.

==Example==
For example, the following matrix is skew-Hermitian
<math display="block"> A = \begin{bmatrix} -i & +2 + i \\ -2 + i & 0 \end{bmatrix}</math>
because
<math display="block">
  -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 matrix|normal]]. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.<ref>{{harvtxt|Horn|Johnson|1985}}, §2.5.2, §2.5.4</ref>
* All entries on the [[main diagonal]] of a skew-Hermitian matrix have to be pure [[imaginary number|imaginary]]; i.e., on the imaginary axis (the number zero is also considered purely imaginary).<ref>{{harvtxt|Meyer|2000}}, Exercise 3.2.5</ref>
* If <math>A</math> and <math>B</math> are skew-Hermitian, then {{tmath|aA + bB}} is skew-Hermitian for all [[real number|real]] [[scalar (mathematics)|scalars]] <math>a</math> and <math>b</math>.<ref name=HJ85S411>{{harvtxt|Horn|Johnson|1985}}, §4.1.1</ref>
* <math>A</math> is skew-Hermitian ''if and only if'' <math>i A</math> (or equivalently, <math>-i A</math>) is [[Hermitian matrix|Hermitian]].<ref name=HJ85S411/>
*<math>A</math> is skew-Hermitian ''if and only if'' the real part <math>\Re{(A)}</math> is [[skew-symmetric matrix|skew-symmetric]] and the imaginary part <math>\Im{(A)}</math> is [[symmetric matrix|symmetric]].
* If <math>A</math> is skew-Hermitian, then <math>A^k</math> is Hermitian if <math>k</math> is an even integer and skew-Hermitian if <math>k</math> is an odd integer.
* <math>A</math> is skew-Hermitian if and only if <math>\mathbf{x}^\mathsf{H} A \mathbf{y} = -\overline{\mathbf{y}^\mathsf{H} A \mathbf{x}}</math> for all vectors <math>\mathbf x, \mathbf y</math>.
* If <math>A</math> is skew-Hermitian, then the [[matrix exponential]] <math>e^A</math> is [[unitary matrix|unitary]].
* The space of skew-Hermitian matrices forms the [[Lie algebra]] <math>u(n)</math> of the [[Lie group]] <math>U(n)</math>.

==Decomposition into Hermitian and skew-Hermitian==
* The sum of a square matrix and its conjugate transpose <math>\left(A + A^\mathsf{H}\right)</math> is Hermitian.
* The difference of a square matrix and its conjugate transpose <math>\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>C</math> can be written as the sum of a Hermitian matrix <math>A</math> and a skew-Hermitian matrix <math>B</math>: <math display="block">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==
* [[Bivector (complex)]]
* [[Hermitian matrix]]
* [[Normal matrix]]
* [[Skew-symmetric matrix]]
* [[Unitary matrix]]

==Notes==
<references/>

==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=[[Society for Industrial and Applied Mathematics|SIAM]] | isbn=978-0-89871-454-8 | year=2000}}.

{{Matrix classes}}

[[Category:Matrices (mathematics)]]
[[Category:Abstract algebra]]
[[Category:Linear algebra]]