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Let $U$ be a finite-dimensional inner-product space over a field $\kfield$ Let $T:U\rightarrow U$ be an endomorphism, and note that the adjoint endomorphism $T\adj$ is also an endomorphism of $U$ It is therefore possible to add, subtract, and compare $T$ and $T\adj$ and we are able to make the following definitions. An
endomorphism $T$ is said to be self-dual (a.k.a. self-adjoint) if $$T=T\adj.$$ By contrast, we say that the endomorphism is anti self-dual if $$T=-T\adj.$$
Exactly the same definitions can be made for an endomorphism of a complex vector space with a Hermitian inner product.
All of these definitions have their counterparts in the matrix setting. Let $M\in \Mat_{n,n}(\kfield)$ 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 matrix, 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 transpose with the conjugate transpose. Thus $T$ is self dual if and only if $M$ is a Hermitian matrix, i.e. $$M = \overline{M^t}.$$ It is anti self-dual if and only if $$M = -\overline{M^t}.$$
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"self-dual" is owned by rmilson.
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Cross-references: conjugate transpose, transpose, skew-symmetric matrix, symmetric matrix, basis, orthogonal, matrix, Hermitian inner product, vector space, complex, definitions, adjoint endomorphism, endomorphism, field, finite-dimensional
There are 11 references to this entry.
This is version 2 of self-dual, born on 2002-02-26, modified 2002-02-26.
Object id is 2719, canonical name is SelfDual.
Accessed 8503 times total.
Classification:
| AMS MSC: | 15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products) | | | 15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices ) | | | 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations) |
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Pending Errata and Addenda
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