228 CHAPTER 8. LINEAR TRANSFORMATIONS

x1

x2

x3

ψ

x3(t)

ϕ

line of nodes

θ

This is as far as I will go on this topic. The point is, it is possible to give a systematicdescription in terms of matrix multiplication of a very elaborate geometrical description ofa composition of linear transformations. You see from the picture it is possible to describethe motion of the spinning top shown in terms of these Euler angles.

8.4 Eigenvalues and Eigenvectors of Linear Transfor-mations

Let V be a finite dimensional vector space. For example, it could be a subspace of Cnor Rn.Also suppose A ∈ L (V, V ) .

Definition 8.4.1 The characteristic polynomial of A is defined as q (λ) ≡ det (λI −A) .The zeros of q (λ) in F are called the eigenvalues of A.

Lemma 8.4.2 When λ is an eigenvalue of A which is also in F, the field of scalars, thenthere exists v ̸= 0 such that Av = λv.

Proof: This follows from Theorem 8.3.16. Since λ ∈ F,

λI −A ∈ L (V, V )

and since it has zero determinant, it is not one to one. ■The following lemma gives the existence of something called the minimal polynomial.

Lemma 8.4.3 Let A ∈ L (V, V ) where V is a finite dimensional vector space of dimensionn with arbitrary field of scalars. Then there exists a unique polynomial of the form

p (λ) = λm + cm−1λm−1 + · · ·+ c1λ+ c0