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


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


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

Relation to the matrix transpose.

All of these definitions have their counterparts in the matrix setting. Let MMatn,n(𝕂) be the matrix of T relative to an orthogonal basis 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 transposeMathworldPlanetmath with the conjugate transposeMathworldPlanetmath. Thus T is self dual if and only if M is a Hermitian matrix, i.e.


It is anti self-dual if and only if

Title self-dual
Canonical name Selfdual
Date of creation 2013-03-22 12:29:40
Last modified on 2013-03-22 12:29:40
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 5
Author rmilson (146)
Entry type Definition
Classification msc 15A63
Classification msc 15A57
Classification msc 15A04
Synonym self-adjoint
Related topic HermitianMatrix
Related topic SymmetricMatrix
Related topic SkewSymmetricMatrix
Defines anti self-dual