Page 231 - 35Linear Algebra

12.1 Invariant Directions 231

Reading homework: problem 2

Now that we’ve seen what eigenvalues and eigenvectors are, there are a
number of questions that need to be answered.

• How do we find eigenvectors and their eigenvalues?

• How many eigenvalues and (independent) eigenvectors does a given
linear transformation have?

• When can a linear transformation be diagonalized?

We will start by trying to find the eigenvectors for a linear transformation.

2 × 2 Example

2
2
Example 127 Let L: R → R such that L(x, y) = (2x + 2y, 16x + 6y). First, we
find the matrix of L, this is quickest in the standard basis:

x L 2 2 x
7−→ .
y 16 6 y

x
We want to find an invariant direction v = such that
y
Lv = λv

or, in matrix notation,

2 2 x x
= λ
16 6 y y

2 2 x λ 0 x
⇔ =
16 6 y 0 λ y

2 − λ 2 x 0
⇔ = .
16 6 − λ y 0
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