Semi-simple operator

In mathematics, a linear operator T on a finite-dimensional vector space is semi-simple if every T-invariant subspace has a complementary T-invariant subspace.

An important result regarding semi-simple operators is that, a linear operator on a finite dimensional vector space over an algebraically closed field is semi-simple if and only if it is diagonalizable. This is because 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

• Kenneth Hoffman and Ray Kunze. "Semi-Simple operators". Linear Algebra. Pearson Education. pp. 262–265. ISBN 81-297-0213-4. 
• Lam, Tsit-Yuen (2001). A first course in noncommutative rings. Graduate texts in mathematics. 131 (2 ed.). Springer. ISBN 0387951830.