Page 346 - 35Linear Algebra
P. 346
346 Sample Final Exam
11. A team of distinguished, post-doctoral engineers analyzes the design for a
bridge across the English channel. They notice that the force on the center
x
of the bridge when it is displaced by an amount X = is given by
y
z
−x − y
F = −x − 2y − z .
−y − z
Moreover, having read Newton’s Principiæ, they know that force is propor-
tional to acceleration so that 2
2
d X
F = .
dt 2
Since the engineers are worried the bridge might start swaying in the heavy
channel winds, they search for an oscillatory solution to this equation of the
form 3
a
X = cos(ωt) .
b
c
(a) By plugging their proposed solution in the above equations the engi-
neers find an eigenvalue problem
a a
b
b
M = −ω 2 .
c c
Here M is a 3 × 3 matrix. Which 3 × 3 matrix M did the engineers
find? Justify your answer.
(b) Find the eigenvalues and eigenvectors of the matrix M.
(c) The number |ω| is often called a characteristic frequency. What char-
acteristic frequencies do you find for the proposed bridge?
(d) Find an orthogonal matrix P such that MP = PD where D is a
diagonal matrix. Be sure to also state your result for D.
2
The bridge is intended for French and English military vehicles, so the exact units,
coordinate system and constant of proportionality are state secrets.
3
Here, a, b, c and ω are constants which we aim to calculate.
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