Page 253 - DMTH502_LINEAR_ALGEBRA

Unit 23: Semi-simple Operators

 It is seen that for a linear operator T on V, a finite dimensional vector space over a field of Notes
complex numbers has a semi-simple operator S on V and a nilpotent operator N on V such
that T = S + N, SN = NS.

23.4 Keywords

Complementary T-invariant subspace: Let T a linear operator has a T-invariant sub-space W
such that V = W W’ then W’ is a subspace which is complementary to W. However if W’ is also
T-invariant then W’ is known as complementary T-invariant subspace.
Semi-simple operator: Let T be a linear operator on V, and suppose that the minimal polynomial
for T is irreducible over the scalar field F, then T is called a semi-simple operator.

23.5 Review Questions

1. Let T be a linear operator on a finite dimensional space over a subfield of C. Prove that T
is semi-simple and only if the following is true. If f is a polynomial and f(T) is nilpotent,
then f(T) = 0.
2. Let T a linear operator on V is represented by the matrix

4 2 2
A = 5 3 2
2 4 1

Show that T is diagonalizable.

23.6 Further Readings

Books Kenneth Hoffman and Ray Kunze, Linear Algebra
Michael Artin, Algebra

LOVELY PROFESSIONAL UNIVERSITY 247