Semi-simple operator

From Wikipedia, the free encyclopedia
  (Redirected from Semisimple operator)
Jump to navigation Jump to search

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

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

References

  • Hoffman, Kenneth; Kunze, Ray (1971). "Semi-Simple operators". Linear algebra (2nd ed.). Englewood Cliffs, N.J.: Prentice-Hall, Inc. MR 0276251.
  • Lam, Tsit-Yuen (2001). A first course in noncommutative rings. Graduate texts in mathematics. 131 (2 ed.). Springer. ISBN 0-387-95183-0.


Retrieved from "https://en.wikipedia.org/w/index.php?title=Semi-simple_operator&oldid=644372405"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Semisimple_operator
This page is based on the copyrighted Wikipedia article "Semi-simple operator"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA