Page 238 - 35Linear Algebra

238 Eigenvalues and Eigenvectors

12.4 Review Problems

Reading Problems 1 , 2 , 3
Characteristic polynomial 4, 5, 6
Eigenvalues 7, 8
Webwork:
Eigenspaces 9, 10
Eigenvectors 11, 12, 13, 14
Complex eigenvalues 15

2
2
1. Try to find more solutions to the vibrating string problem ∂ y/∂t =
2
2
∂ y/∂x using the ansatz
y(x, t) = sin(ωt)f(x) .
What equation must f(x) obey? Can you write this as an eigenvector
equation? Suppose that the string has length L and f(0) = f(L) = 0.
Can you find any solutions for f(x)?

2 1
2. Let M = . Find all eigenvalues of M. Does M have two linearly
0 2
independent eigenvectors? Is there a basis in which the matrix of M is
diagonal? (I.e., can M be diagonalized?)

2
2
3. Consider L: R → R with

x x cos θ + y sin θ
L = .
y −x sin θ + y cos θ

1 0
(a) Write the matrix of L in the basis , .
0 1
(b) When θ 6= 0, explain how L acts on the plane. Draw a picture.
(c) Do you expect L to have invariant directions? (Consider also
special values of θ.)
(d) Try to find real eigenvalues for L by solving the equation

L(v) = λv.
√
(e) Are there complex eigenvalues for L, assuming that i = −1
exists?

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