Page 205 - DMTH502_LINEAR_ALGEBRA

Sachin Kaushal, Lovely Professional University Unit 18: The Primary Decomposition Theorem

Unit 18: The Primary Decomposition Theorem Notes

CONTENTS
Objectives
Introduction

18.1 Overview
18.2 Primary Decomposition Theorem
18.3 Summary

18.4 Keywords
18.5 Review Questions
18.6 Further Readings

Objectives

After studying this unit, you will be able to:

 See that in considering a linear operator T on a finite dimensional space the minimal
polynomial for the linear operator is a product of a number of irreducible monic
i r
polynomials p over the field F where r are positive integers.
i i
 Know that this structure of the minimal polynomial helps in decomposing the space V as
the direct sum of the invariant subspaces W .
i
 Understand that the general linear operator T induces a linear operator T on W by restriction
i i
i r
and the minimal polynomial for T is the irreducible p .
i i
Introduction

In this unit the idea of the direct sum decomposition of the vector space V for a linear operator
T in terms of invariant subspaces.
The general linear operator T induces a linear operator T on the invariant subspace, the minimal
i
i r
polynomial of T is the p .
i i
This structure of the induced linear operator helps in introducing the projection operators E .
i
These projections associated with the primary decomposition of T, then are polynomials in T,
and they commute each will an operator that commutes with T.

18.1 Overview

We continue our study of a linear operator T on the finite dimension space. In this unit we are
interested in decomposing T into a direct sum of operators which are in some sense elementary.
We had already found the characteristic values of the operator and also studied invariant
subspaces. The vector space V was shown to be direct sum of the invariant subspaces. We can
decompose T into a direct sum of operators through the characteristic values and vectors of T in
certain special cases i.e., when the minimal polynomial for T factors over the scalar field F into
a product of distinct monic polynomials of degree 1. In dealing with the general T we come

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