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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 x : V → V {\displaystyle x:V\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.