# self-dual

## Definition.

Let $U$ 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 $U$. It is therefore possible to add, subtract, and compare $T$ and $T^{\displaystyle\star}$, and we are able to make the following definitions. An endomorphism $T$ is said to be self-dual (a.k.a. ) if

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

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

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

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 $T$ 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

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