18.3. EIGENVALUES AND EIGENVECTORS OF LINEAR TRANSFORMATIONS431
Proof: Consider the linear transformations, I,A,A2, · · · ,An2. There are n2 +1 of these
transformations and so by Theorem 18.2.4 the set is linearly dependent. Thus there existconstants, ci ∈ C such that
c0I +n2
∑k=1
ckAk = 0.
This implies there exists a polynomial, q(λ ) which has the property that q(A) = 0. In fact,q(λ )≡ c0 +∑
n2
k=1 ckλk. Dividing by the leading term, it can be assumed this polynomial is
of the form λm + cm−1λ
m−1 + · · ·+ c1λ + c0, a monic polynomial. Now consider all suchmonic polynomials q such that q(A) = 0 and pick one which has the smallest degree. Thisis called the minimal polynomial and will be denoted here by p(λ ) . By the fundamentaltheorem of algebra, p(λ ) is of the form
p(λ ) =m
∏k=1
(λ −λ k) .
where some of the λ k might be repeated. Thus, since p has minimal degree,
m
∏k=1
(A−λ kI) = 0, butm−1
∏k=1
(A−λ kI) ̸= 0.
Therefore, there exists u ̸= 0 such that
v≡
(m−1
∏k=1
(A−λ kI)
)(u) ̸= 0.
But then
(A−λ mI)v = (A−λ mI)
(m−1
∏k=1
(A−λ kI)
)(u) = 0. ■
As a corollary, it is good to mention that the minimal polynomial just discussed isunique.
Corollary 18.3.7 Let A ∈ L (V,V ) where V is an n dimensional vector space, the fieldof scalars being F. Then there exists a polynomial q(λ ) having coefficients in F suchthat q(A) = 0. Letting p(λ ) be the monic polynomial having smallest degree such thatp(A) = 0, it follows that p(λ ) is unique.
Proof: The existence of p(λ ) follows from the above theorem. Suppose then thatp1 (λ ) is another one. That is, it has minimal degree of all polynomials q(λ ) satisfyingq(A) = 0 and is monic. Then by Lemma 16.4.3 there exists r (λ ) which is either equal to 0or has degree smaller than that of p(λ ) and a polynomial l (λ ) such that
p1 (λ ) = p(λ ) l (λ )+ r (λ )
By assumption, r (A) = 0. Therefore, r (λ ) = 0. Also by assumption, p1 (λ ) and p(λ ) havethe same degree and so l (λ ) is a scalar. Since p1 (λ ) and p(λ ) are both monic, it followsthis scalar must equal 1. This shows uniqueness. ■