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单词 Selfdual
释义

self-dual


Definition.

Let U be a finite-dimensional inner-product spaceover a field 𝕂. Let T:UU be an endomorphism,and note that the adjoint endomorphism T is also an endomorphismof U. It is therefore possible to add, subtract, and compare Tand T, and we are able to make the following definitions. Anendomorphism T is said to be self-dual (a.k.a. self-adjointPlanetmathPlanetmath) if

T=T.

By contrast, we say that the endomorphism is anti self-dual if

T=-T.

Exactly the same definitions can be made for an endomorphism ofa complex vector space with a Hermitian inner productMathworldPlanetmath.

Relation to the matrix transpose.

All of these definitions havetheir counterparts in the matrix setting. Let MMatn,n(𝕂) be the matrix of T relative to an orthogonalbasis of U. Then T is self-dual if and only if M is a symmetric matrixMathworldPlanetmath,and anti self-dual if and only if M is a skew-symmetric matrix.

In the case of a Hermitian inner product we must replace the transposeMathworldPlanetmathwith the conjugate transposeMathworldPlanetmath. Thus T is self dual if and only if M is a Hermitian matrix, i.e.

M=Mt¯.

It is anti self-dual if and only if

M=-Mt¯.
Titleself-dual
Canonical nameSelfdual
Date of creation2013-03-22 12:29:40
Last modified on2013-03-22 12:29:40
Ownerrmilson (146)
Last modified byrmilson (146)
Numerical id5
Authorrmilson (146)
Entry typeDefinition
Classificationmsc 15A63
Classificationmsc 15A57
Classificationmsc 15A04
Synonymself-adjoint
Related topicHermitianMatrix
Related topicSymmetricMatrix
Related topicSkewSymmetricMatrix
Definesanti self-dual
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更新时间:2025/5/4 10:58:05