Page 173 - DMTH502_LINEAR_ALGEBRA

Richa Nandra, Lovely Professional University Unit 13: Annihilating Polynomials

Unit 13: Annihilating Polynomials Notes

CONTENTS
Objectives
Introduction

13.1 Overview
13.2 Annihilating Polynomials
13.3 Summary

13.4 Keywords
13.5 Review Questions
13.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Know about the polynomials over the field F, the degree of polynomial, monic polynomial,
annihilating polynomials as well as minimal polynomials.
 Understand that the linear operator is annihilated by its characteristic polynomial.

 Understand that we consider all monic polynomials with coefficients in F and the degree
of the minimal polynomial is the least positive integer such that a linear relation is
obtained annihilated.
Introduction

In this unit we investigate more properties of a linear transformation.
We define certain terms like monic polynomial, minimal polynomial as well as annihilating
polynomial and characteristic polynomial.
It is seen that the theorem of Cayley-Hamilton in this unit helps us in narrowing down the reach
for the minimal polynomials of various operators.

13.1 Overview

2
n
Polynomial Over F. Let F(x) be the subspace of F spanned by vectors 1, x, x ..... An element of
F(x) is called a polynomial over F.
Degree of a Polynomial: F(x) consists of all (finite) linear combinations of x and its powers. If f is
a non-zero polynomial of the form
f f x 0 f x f x 2  f x n
0
n
1
2
such that f n 0 and n 0 and f n 0 for all integers k > n; this integer is obviously unique and is
called the degree of f.
The scalars f , f , f , , f are sometimes called the coefficients of f in the field F.
0 1 2 n

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