Page 181 - DMTH502_LINEAR_ALGEBRA

Unit 13: Annihilating Polynomials

Self Assessment Notes

1. Let A be the following 3 3 matrix over F;

2 1 1
A = 2 2 1
1 1 2

Find the characteristic polynomial for A and also the minimal polynomial for A.

2. Let A be the following 3 3 matrix over F;

1 3 7
A = 4 2 3
1 2 1

Find the characteristic polynomial for A and also find the minimal polynomial for A.

13.3 Summary

 In this unit certain terms related to linear operator T are defined, i.e., the monic polynomial,
annihilating polynomials, minimal polynomials as well as characteristic polynomials.

 With the help of Cayley-Hamilton theorem it becomes easier to search for the minimal
polynomials of various operators.

13.4 Keywords

Annihilating Polynomial: Annihilating polynomial f(x) over the field F is such that for a matrix
A of n n matrix over the field f(A) = 0, then we say that the polynomial annihilates the matrix.
If a linear operator T is represented by the matrix then f(T) = 0 gives us the annihilating polynomial
for the linear operator T.
Monic Polynomial: The monic polynomial is a polynomial f(x) whose coefficient of the highest
degree in it is unity.

13.5 Review Questions

1. Let A be the following 3 3 matrix over F;

2 4 3
A = 0 1 1
2 2 1

Find the characteristic polynomial and minimal polynomial for A.

2. Let A be the following 3 3 matrix over F;

1 2 0
A = 2 1 0
0 0 1

Find the characteristic polynomial and minimal polynomial for A.

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