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. ■