PHZ 3113 Fall 2010
Homework #8, Due Friday, November 19
1. Let
A=
1 α2
α α3
.
What are the eigenvalues of A? What are the eigenvectors of A? For what α are the
eigenvectors orthogonal?
2. Let
P =
0
0
1
0 −1
1
0
0
0
!
,
1
Q= √
2
0 1 0
1 0 1
0 1 0
!
.
(a) Which of P , Q has real eigenvalues? (Cite a reason that does not involve computing the
eigenvalues.) Which is invertible? Which is a rotation? For the one that is invertible, find
the inverse.
(b) Find the eigenvalues and eigenvectors for the matrix that has real eigenvalues.
3. Three equal masses m are connected to each other along one line by springs with spring
constant k and to walls by springs with spring constant K, such that the departures of the
three masses from their equilibrium points obey
m ẍ1 = −Kx1 − k(x1 − x2 ),
m ẍ2 = −k[(x2 − x1 ) + (x2 − x3 )],
m ẍ3 = −k(x3 − x2 ) − Kx3 .
(a) Assume there are oscillatory solutions, xi = ai cos ωt. Show that this leads to a set of
algebraic (not differential) equations. Write the left- and right-hand sides of the algebraic
equations in matrix form.
(b) What are the characteristic frequencies of oscillations? What are the oscillation mode
patterns?
(c) Examine your frequencies and modes for K → 0. Comment. Examine your frequencies
and modes for k → 0. Comment.