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Sachin Kaushal, Lovely Professional University Unit 23: Semi-simple Operators

Unit 23: Semi-simple Operators Notes

CONTENTS
Objectives
Introduction

23.1 Overview
23.2 Semi-simple Operators
23.3 Summary

23.4 Keywords
23.5 Review Questions
23.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the meaning of semi-simple linear operator T by means of a few lemma stated
in this unit.
 See that if T is a linear operator on V and the minimal polynomial for T is irreducible over
the scalar field then T is semi-simple.
 Know that T, a linear operator on a finite-dimensional space is semi-simple if and only if
T is diagonalizable.
 Understand that if T is a linear operator on V, a finite dimensional vector space over F a
subfield of the field of complex numbers, then there is a semi-simple operator S and a
nilpotent operator N on V such that T = S + N and SN = NS.

Introduction

In this unit the outcome of the last few units is reviewed and a few lemmas based on these ideas
are proved.
The criteria for an operator to be semi-simple are given. It is shown that a linear operator on
finite dimensional space having minimal polynomial to be irreducible is semi-simple.

It is also shown that for a linear operator T on a finite dimensional vector space V over the field
F which is subfield of the field of complex numbers, the operator is the sum of a semi-simple
operator S on V and a nilpotent operator N on V such that T = S + N and SN = NS.
23.1 Overview

In the last couple of units we have been dealing with a single linear operator T on a finite
dimensional vector space V. The aim has been to decompose T into a direct sum of linear
operators of an elementary nature.
We first of all studied the characteristic values and characteristic vectors and also constructed
diagonalizable operators. It was observed then that the characteristic vectors of T need not space
the space.

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