Semi-simple operator

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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.[1]

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.[1] 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[edit]

  1. ^ a b Lam (2001), p. 39

References[edit]

  • 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 0-387-95183-0.