Page 217 - DMTH502_LINEAR_ALGEBRA

Richa Nandra, Lovely Professional University Unit 20: Cyclic Decomposition and the Rational Form

Unit 20: Cyclic Decomposition and the Rational Form Notes

CONTENTS
Objectives
Introduction

20.1 Overview
20.2 Cyclic Decomposition
20.3 The Rational Form

20.4 Summary
20.5 Keywords
20.6 Review Questions
20.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand that if T is any linear operator on a finite dimensional space V, then there
exists vectors , , ... in V such that the space V is a direct sum of the T-cyclic subspaces
1 2 n
Z( ; T) for i = 1, 2, ... n.
i
 See that if W is any subspace of V, then there exists a subspace W , called complementary to
W, such that V = W W .
 Know that if W is T-invariant and W complementary to W is also T-invariant then W is
also T-admissible.

 Understand that the Cyclic decomposition theorem says that there exist non-zero vectors
, , ... in V with respective T-annihilators p , p , ... p such that, V is a direct sum of
1 2 n 1 2 r
T-invariant subspaces along with a proper T-admissible subspace W.

Introduction

In this unit certain concepts like invariant cyclic subspaces, complimentary subspaces and
T-admissible proper subspaces are introduced.
The Cyclic decomposition theorem helps us in decomposing the n-dimensional vector space as
a direct sum of T-invariant cyclic subspaces.
The matrix analogue of the Cyclic Decomposition theorem is that for the cyclic ordered basis
( , T , 2 , ... T k–1 ) the matrix of induced operator T is the companion matrix Ai if the polynomial
i
p , the matrix
i
A 1 0 ... 0
0 A 2 ... 0
A
  ... 0
0 0 ... A n

having a rational form.

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