Editing Skew-Hermitian matrix (section)

{{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.