Page 236 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Richa Nandra, Lovely Professional University

Notes Unit 22: Computation of Invariant Factors

CONTENTS
Objectives
Introduction

22.1 Overview
22.2 Computation of Invariant Factors
22.3 Summary

22.4 Keywords
22.5 Review Question
22.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand how to obtain the characteristic polynomial for a matrix of large size with the
help of the elementary row and column operations.
 See that this unit gives a detailed method which can be used by computation of invariant
factors as the matrix involved depends upon the polynomials in the field F (x).
n
 See that with method of elementary row and column operations a matrix can be put into
the Jordan form.
 Understand that if P is an m × m matrix with entries in the polynomial algebra F(x) then P
is invertible means that P is row equivalent to the m × m identity matrix and P is a product
of elementary matrices.

Introduction

In this unit a method for computing the invariant factors p , ... p is given where p , p , ... p define
1 r 1 2 r
the rational form for the n n matrix A.
The elementary row operations and column operations are to be used to reduce (xI – A) into an
row equivalent matrix.

It is also shown that if N is row equivalent to M then N = PM, where P an m m matrix is a product
of elementary matrices.

22.1 Overview

We wish to find a method for computing the invariant factors p , p , ... p which define the
1 2 r
rational form for an n n matrix A with entries in the field F. To begin with a very simple case
in which A is the companion matrix (2) of unit 9 of a monic polynomial

n
p = x + C x + ... + C x + C .
n–1
n–1 1 0
In unit (19) we saw that p is both the minimal and the characteristic polynomial for the companion
matrix A. Now, we want to give a direct calculation which shows that p is the characteristic
polynomial for A.

230 LOVELY PROFESSIONAL UNIVERSITY