Page 214 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes between the vectors T , that is, we shall be interested in the polynomials g = c + c x + ... + c x k
j
0 1 k
which have the property that g(T) = 0. The set of all g in F[x] such that g(T) = 0 is clearly an ideal
in F[x]. It is also a non-zero ideal, because it contains the minimal polynomial p of the operator
T(p)(T) = 0 for every  in V).
Definition: If  is any vector in V, the T-annihilator of  is the ideal M(; T) in F[x] consisting of
all polynomials g over F such that g(T) = 0. The unique monic polynomial p which generates

this ideal will also be called the T-annihilator of .
As we pointed out above, the T-annihilator p divides the minimal polynomial of the

operator T. Please note that deg(p ) > 0 unless  is the zero vector.

Theorem 1: Let  be any non-zero vector in V and let p be the T-annihilator of .

(i) The degree of p is equal to the dimension of the cyclic subspace Z(; T).

k–1
(ii) If the degree of p is k, then the vectors , T, T , ..., T  form a basis for Z(; T).
2

(iii) If U is the linear operator on Z(; T) induced by T, then the minimal polynomial for U
is p .

Proof: Let g be any polynomial over the field F. Write
g = p q + r

where either r = 0 or deg (r) < deg (p ) = k. The polynomial p q is in the T-annihilator of , and so
 
g(T) = r(T)
k–1
Since r = 0 or deg (r) < k, the vector r(T) is a linear combination of the vectors , T, ..., T , and
since g(T) is a typical vector in Z(; T), this shows that these k vectors span Z(; T). These vectors
are certainly linearly independent, because any non-trivial linear relation between them would
give us a non-zero polynomial g such that g(T) = 0 and deg(g) < deg(p ), which is absurd. This

proves (i) and (ii).
Let U be the linear operator on Z(; T) obtained by restricting T to that subspace. If g is any
polynomial over F, then
p (U)g(T) = p (U)g(T)
 
= g(T)p (U)

= g(T)0

= 0
Thus the operator p (U) sends every vector in Z(; T) into 0 and is the zero operator on Z(, T).

Furthermore, if h is a polynomial of degree less than k, we cannot have h(U) = 0, for then h(U) =
h(T) = 0, contradicting the definition of p . This shows that p is the minimal polynomial for U.
 
A particular consequence of this theorem is the following: If  happens to be a cyclic vector for
T, then the minimal polynomial for T must have degree equal to the dimension of the space V;
hence, the Cayley-Hamilton theorem tells us that the minimal polynomial for T is the
characteristic polynomial for T. We shall prove later that for any T there is a vector  in V which
has the minimal polynomial of T for its annihilator. It will then follow that T has a cyclic vector
if and only if the minimal and characteristic polynomials for T are identical. But it will take a
little work for us to see this.
Our plan is to study the general T by using operators which have a cyclic vector. So, let us take a
look at a linear operator U on a space W of dimension k which has a cyclic vector . By Theorem 1,
k–1
the vectors , ..., U  form a basis for the space W, and the annihilator p of  is the minimal


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