Page 240 - 35Linear Algebra

240 Eigenvalues and Eigenvectors

What do you find from this computation? Does something similar hold
for 3 × 3 matrices? (Try assuming that the matrix of M is diagonal to
answer this.)

9. Discrete dynamical system. Let M be the matrix given by

3 2
M = .
2 3

x(0)
Given any vector v(0) = , we can create an infinite sequence of
y(0)
vectors v(1), v(2), v(3), and so on using the rule:

v(t + 1) = Mv(t) for all natural numbers t.

(This is known as a discrete dynamical system whose initial condition
is v(0).)

(a) Find all eigenvectors and eigenvalues of M.

(b) Find all vectors v(0) such that

v(0) = v(1) = v(2) = v(3) = · · ·

(Such a vector is known as a fixed point of the dynamical system.)

(c) Find all vectors v(0) such that v(0), v(1), v(2), v(3), . . . all point in
the same direction. (Any such vector describes an invariant curve
of the dynamical system.)

Hint

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