self-dual

Definition.

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

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

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

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

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 $M\in {Mat}_{n,n}(𝕂)$ 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 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