Linear Algebra, Theory And Applications, 2012a

240 LINEAR TRANSFORMATIONS
linear transformation which is obtained as a composition of the three just described? By
Theorem 9.3.17, this matrix equals the product of these three,
M 3 (φ, θ, ψ) M 2 (φ, θ) M 1 (φ) .
I leave the details to you. There are procedures due to Lagrange which will allow you to
write differential equations for the Euler angles in a rotating body. To give an idea how
these angles apply, consider the following picture.
x 3
x 3 (t)
❘ ψ
θ
x 2
φ
x 1
line of nodes
This is as far as I will go on this topic. The point is, it is possible to give a systematic
description in terms of matrix multiplication of a very elaborate geometrical description of
a composition of linear transformations. You see from the picture it is possible to describe
the motion of the spinning top shown in terms of these Euler angles.
9.4 Eigenvalues And Eigenvectors Of Linear Transformations
Let V be a finite dimensional vector space. For example, it could be a subspace of C n or R n .
Also suppose A ∈L(V,V ) .
Definition 9.4.1 The characteristic polynomial of A is defined as q (λ) ≡ det (λI − A) .
The zeros of q (λ) in C are called the eigenvalues of A.
Lemma 9.4.2 When λ is an eigenvalue of A whichisalsoinF, the field of scalars, then
there exists v ≠0such that Av = λv.
Proof: This follows from Theorem 9.3.16. 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 9.4.3 Let A ∈L(V,V ) where V is a finite dimensional vector space of dimension
n with arbitrary field of scalars. Then there exists a unique polynomial of the form
p (λ) =λ m + c m−1 λ m−1 + ···+ c 1 λ + c 0