PlanetMath: self-dual

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self-dual

(Definition)

Definition.

Let

be a finite-dimensional inner-product space over a field $\mathbb{K}$ . Let $T:U\rightarrow U$ be an endomorphism, and note that the adjoint endomorphism $T^{\displaystyle \star}$ is also an endomorphism of

. It is therefore possible to add, subtract, and compare

and $T^{\displaystyle \star}$ , and we are able to make the following definitions. An endomorphism

is said to be self-dual (a.k.a. self-adjoint) if

$\displaystyle T=T^{\displaystyle \star}.$

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

$\displaystyle T=-T^{\displaystyle \star}.$

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

Relation to the matrix transpose.

All of these definitions have their counterparts in the matrix setting. Let $M\in \mathop{\mathrm{Mat}}\nolimits _{n,n}(\mathbb{K})$ be the matrix of

relative to an orthogonal basis of

. Then

is self-dual if and only if

is a symmetric matrix, and anti self-dual if and only if

is a skew-symmetric matrix.

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

$\displaystyle M = \overline{M^t}.$

It is anti self-dual if and only if

$\displaystyle M = -\overline{M^t}.$

"self-dual" is owned by rmilson.

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Other names:

self-adjoint

Also defines:

anti self-dual

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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 10 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 6791 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)

Pending Errata and Addenda

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