Page 174 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Monic Polynomial: A polynomial f(x) over a field F is called monic polynomial if the coefficient
of highest degree term in it is unity i.e f 0
n
Annihilating Polynomials: Let A be n n matrix over a field F and f(x) be a polynomial over F.
Then if f(A) = 0. Then we say that the polynomial f(x) annihilates the matrix A.

13.2 Annihilating Polynomials

It is important to know the class of polynomials that Annihilate T.
Suppose T is a linear operator on V, a vector space over the field F. If p is a polynomial over F,
then p(T) is again a linear operator on V. If q is another polynomial over F, then

p
T
T
(p q )( ) = ( ) q ( )
T
p
T
T
T
q
(pq )( ) = ( ) ( )
Therefore, the collection of polynomials p which annihilate T, in the sense that
T
p ( ) = 0,
is an ideal in the polynomial algebra F[x]. It may be the zero ideal, i.e., it may be that T is not
annihilated by any non-zero polynomial. But, that cannot happen if the space V is finite-
dimensional.
2
Suppose T is a linear operator on the n-dimensional space V. Look at the first (n + 1) powers of T:
2
I , ,T  ,T n 2 .
T
This is a sequence of n + 1 operators in L(V, V), the space of linear operators on V. The space
2
2
2
L(V, V,) has dimension n . Therefore, that sequence of n + 1 operators must be linearly dependent.
i.e., we have
c I c T  c T n 2 0
0
n
2
1
for some scalars c not all zero. So, the ideal of polynomials which annihilate T contains a non-
i
zero polynomial of degree n or less.
2
Definition. Let T be a linear operator on a finite-dimensional vector space V over the field F. The
minimal polynomial for T is the (unique) monic generator of the ideal of polynomials over F
which annihilate T.
The name ‘minimal polynomial’ stems from the fact that generator of a polynomial ideal is
characterized by being the monic polynomial of minimum degree in the ideal. That means that
the minimal polynomial p for the linear operator T is uniquely determined by these three
properties:
1. p is a monic polynomial over the scalar field F.
2. p(T) = 0

3. No polynomial over F which annihilates T has smaller degree than p has.
If A an n n matrix over F, we define the minimal polynomial for A in an analogous way, as the
unique monic generator of the ideal of all polynomials over F which annihilate A. If the operator
T is represented in some ordered basis by the matrix A, then T and A have the same minimal
polynomial. That is because f(T) is represented in the basis by the matrix f(A) so that f(T) = 0 if and
only if f(A) = 0.

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