Semisimple operator

{{Short description|Linear operator}}

In mathematics, a linear operator T : V → V on a vector space V is semisimple if every T-invariant subspace has a complementary T-invariant subspace.Lam (2001), [{{Google books|plainurl=y|id=f15FyZuZ3-4C|page=39|text=linear operator}} p. 39] If T is a semisimple linear operator on V, then V is a semisimple representation of T. Equivalently, a linear operator is semisimple if its minimal polynomial is a product of distinct irreducible polynomials.{{harvnb|Jacobson|1979|loc=A paragraph before Ch. II, § 5, Theorem 11.}}

A linear operator on a finite-dimensional vector space over an algebraically closed field is semisimple if and only if it is diagonalizable.This is trivial by the definition in terms of a minimal polynomial but can be seen more directly as follows. Such an operator always has an eigenvector; if it is, in addition, semi-simple, then it has a complementary invariant hyperplane, which itself has an eigenvector, and thus by induction is diagonalizable. Conversely, diagonalizable operators are easily seen to be semi-simple, as invariant subspaces are direct sums of eigenspaces, and any basis for this space can be extended to an eigenbasis.

Over a perfect field, the Jordan–Chevalley decomposition expresses an endomorphism $x\; :\; V\; \backslash to\; V$ as a sum of a semisimple endomorphism s and a nilpotent endomorphism n such that both s and n are polynomials in x.