Semisimple operator explained

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.^[1] 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.

A linear operator on a finite-dimensional vector space over an algebraically closed field is semisimple if and only if it is diagonalizable.^{[1] [2]}

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.

See also

• Jordan–Chevalley decomposition

Notes

1. Lam (2001), [{{Google books|plainurl=y|id=f15FyZuZ3-4C|page=39|text=linear operator}} p. 39]
2. 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.

References

• Book: Hoffman . Kenneth . Kunze . Ray . Ray Kunze . Semi-Simple operators . 2nd . Englewood Cliffs, N.J. . 0276251 . Prentice-Hall, Inc. . Linear algebra . 1971.
• Book: Jacobson, Nathan . Nathan Jacobson . Lie algebras . New York . 1979 . 0-486-63832-4 . 6499793 .
• Book: Lam . Tsit-Yuen . A first course in noncommutative rings . 2 . Graduate texts in mathematics . 131 . 2001 . Springer . 0-387-95183-0 .