Page 228 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Richa Nandra, Lovely Professional University

Notes Unit 21: The Jordan Form

CONTENTS
Objectives
Introduction

21.1 Overview
21.2 Jordan Form
21.3 Summary

21.4 Keywords
21.5 Review Questions
21.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the finite vector space V for a linear operator T can be written as a direct sum
of the cyclic invariant subspaces Z(, T).
i
 Know that the characteristic polynomial f of T decomposes as the product of the individual
characteristic polynomial p = x for the r annihilators such that k  k  ...  k . The minimal
ki
i 1 2 r
polynomial also has the form
r
p = (x – c ) 1 ... (x – c ) k r
1 k
 See that with the help of the companion matrix the linear operator represented by the
matrix can be put into the Jordan form.
Introduction

In this unit the findings of the unit 20 are used to put any matrix A representing the linear
operator into the Jordan form.
It is seen that by using the idea of the direct decomposition of the vector space into the sum of the
cyclic subspaces the given matrix A can be shown to be similar to a Jordan matrix.

21.1 Overview

Suppose that N is a nilpotent linear operator on a finite-dimensional space. From Theorem 1 of
the last unit we find that with N-annihilators p , p , ..., p for r non-zero vectors  ,  ,..., , V is
1 2 r 1 2 r
decomposed as follows:
...
V = Z( , N) Z( , N)
1 r
K
Here p divides for i = 1,..., r–1. As N is nilpotent the minimal polynomial is x for K  n, thus
i+1
each p = x , such that
ki
i
K = K  K  ... K
1 1 2 r

222 LOVELY PROFESSIONAL UNIVERSITY