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| In [[linear algebra]], a [[square matrix]] with [[Complex number|complex]] entries is said to be '''skew-Hermitian''' or '''antihermitian''' if its [[conjugate transpose]] is equal to its negative.<ref>{{harvtxt|Horn|Johnson|1985}}, §4.1.1; {{harvtxt|Meyer|2000}}, §3.2</ref> That is, the matrix ''A'' is skew-Hermitian if it satisfies the relation
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| :<math>A^\dagger = -A,\;</math>
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| where <math>\dagger</math> denotes the conjugate transpose of a matrix. In component form, this means that
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| :<math>a_{i,j} = -\overline{a_{j,i}},</math>
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| for all ''i'' and ''j'', where ''a''<sub>''i'',''j''</sub> is the ''i'',''j''-th entry of ''A'', and the overline denotes [[complex conjugate|complex conjugation]].
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| 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> All skew-Hermitian <var>n</var>×<var>n</var> matrices form the '''u'''(<var>n</var>) [[Lie algebra]], which corresponds to the Lie group [[Unitary group|U(<var>n</var>)]].
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| The concept can be generalized to include [[linear transformation]]s of any [[complex number | complex]] [[vector space]] with a [[sesquilinear]] [[Norm (mathematics)|norm]].
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| == Example ==
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| For example, the following matrix is skew-Hermitian:
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| :<math>\begin{bmatrix} -i & 2 + i \\ -(2 - i) & 0 \end{bmatrix}</math>
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| == Properties ==
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| * The eigenvalues of a skew-Hermitian matrix are all purely imaginary or 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>
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| * 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>
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| * If ''A, B'' are skew-Hermitian, then ''aA + bB'' is skew-Hermitian for all [[real number|real]] [[scalar (mathematics)|scalars]] ''a'' and ''b''.<ref name=HJ85S411>{{harvtxt|Horn|Johnson|1985}}, §4.1.1</ref>
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| * If ''A'' is skew-Hermitian, then both ''i A'' and −''i A'' are [[Hermitian matrix|Hermitian]].<ref name=HJ85S411/>
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| * If ''A'' is skew-Hermitian, then ''A''<sup>''k''</sup> is Hermitian if ''k'' is an even integer and skew-Hermitian if ''k'' is an odd integer.
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| * An arbitrary (square) matrix ''C'' can uniquely be written as the sum of a Hermitian matrix ''A'' and a skew-Hermitian matrix ''B'':<ref name=HJ85S412/>
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| ::<math>C = A+B \quad\mbox{with}\quad A = \frac{1}{2}(C + C^\dagger) \quad\mbox{and}\quad B = \frac{1}{2}(C - C^\dagger).</math>
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| * If ''A'' is skew-Hermitian, then e<sup>''A''</sup> is [[unitary matrix|unitary]].
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| * The space of skew-Hermitian matrices forms the [[Lie algebra]] u(''n'') of the [[Lie group]] U(''n'').
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| ==See also==
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| *[[Bivector (complex)]]
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| *[[Hermitian matrix]]
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| *[[Normal matrix]]
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| *[[Skew-symmetric matrix]]
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| *[[Unitary matrix]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{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}}.
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| * {{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}}.
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| [[Category:Matrices]]
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