Let be a finite-dimensional inner-product space over a field . Let be an endomorphism, and note that the adjoint endomorphism is also an endomorphism of . It is therefore possible to add, subtract, and compare and , and we are able to make the following definitions. An endomorphism is said to be self-dual (a.k.a. self-adjoint) if
By contrast, we say that the endomorphism is anti self-dual if
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 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.
It is anti self-dual if and only if
|Date of creation||2013-03-22 12:29:40|
|Last modified on||2013-03-22 12:29:40|
|Last modified by||rmilson (146)|