Semisimple operator

In mathematics, a linear operator "T" on a vector space is semisimple if every "T"-invariant subspace has a complementary "T"-invariant subspace; in other words, the vector space is a semisimple representation of the operator "T". Equivalently, a linear operator is semisimple if the minimal polynomial of it is a product of distinct irreducible polynomials.
A linear operator on a finite dimensional vector space over an algebraically closed field is semisimple if and only if it is diagonalizable.
Over a perfect field, the Jordan–Chevalley decomposition expresses an endomorphism formula_1 as a sum of a semisimple endomorphism "s" and a nilpotent endomorphism "n" such that both "s" and "n" are polynomials in "x".