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semisimple 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.〔Lam (2001), (p. 39 )〕
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.
==Notes==
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