self-dual (original) (raw)

Definition.

Let U be a finite-dimensional inner-product space over a field 𝕂. Let T:U→U 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 Tand T⋆, and we are able to make the following definitions. An endomorphism T is said to be self-dual (a.k.a. self-adjoint) if

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

Relation to the matrix transpose.

All of these definitions have their counterparts in the matrix setting. Let M∈Matn,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 transposewith the conjugate transpose. 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