Canonical Forms Linear Algebra Notes

3 Annihilating Polynomials
Suppose K is a commutative ring and M be a K−module. For
x ∈ M, we define annihiltor of x as
ann(x) = {λ ∈ K : λx = 0}.
Note that ann(x) is an ideal of K. (That means
ann(x) + ann(x) ⊆ ann(x) and K ∗ ann(x) ⊆ ann(x)).
We shall consider annihilator of a linear operator, as follows.
3.1 Minimal (monic) polynomials
3.1 (Facts) Let V be a vector space over a field F with dim(V ) =
n.
Recall, we have seen that M = L(V, V ) is a F[X]−module. For
f(X) ∈ F[X] and T ∈ L(V, V ), scalar multiplication is defined by
f ∗ T = f(T ) ∈ L(V, V )
1. So, for a linear operator T ∈ L(V, V ), the annihilator of T is:
ann(T ) = {f(X) ∈ F[X] : f(T ) = 0}
is an ideal of the polynomial ring F[X].
2. Note that ann(T ) is a non-zero proper ideal. It is non-zero
because dim(L(V, V )) = n 2 and hance
is a linearly dependent set.
1, T, T 2 , . . . , T n2
3. Also recall that any ideal I of F[X] is a principal ideal, which
means that I = F[X]p where p is the non-zero monic in I
polynomial of least degree.
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