18.034 Problem Set #8
(modified on April 27, 2009)
Due by Friday, May 1, 2009, by NOON
1. An n × n complex matrix A is called Hermitian if A = A∗, where A∗ is the conjugate transpose
of A. That is, aij = aji for all 1 ≤ i, j ≤ n. When A is real A∗ = AT and the terms “Hermitian” and
”symmetric” mean the same thing. If �u = (u1, . . . , un) and �v = (v1, . . . , vn) are column vectors in
Rn, then �u�v∗ = u1v¯1 + + unv¯n and �u�2 = uu∗.· · ·
If A = A∗ show that all eienvalues of A are real. Furthermore, if A = A∗ then eigenvectors �u
and �v corresponding to different eigenvalues λ and µ are orthogonal. That is, �u�v∗ = 0.
2. If A and B are n × n matrices, compute
At Bt Bt At
lim e e − e e .
t0 t2
→
3. (a) Let A be a 3 × 3 matrix with eigenvalues λ1, λ2, λ3. Show that the nonzero columns of
(A − λ2I)(A − λ3I) are eigenvectors for λ1.
(b) A 3 × 3 matrix A has characteristic polynomial p(λ) = λ(λ2 − 1). Find eAt
.
4. (a) For x� = 6x + y, y� = 4x + 3y show that the origin is an unstable node.
(b) If y = mx is a trajectory, show that m = 1 or m = −4.
(c) Sketch the trajectories in the (x, y)-plane.
5. Repeat Problem 4 for x� = −3x + 2y, y� = −3x + 4y.
6. Consider the differential equation u�� + p(t)u� + q(t)u = 0, where p(t), q(t) are continuous func-
tions on some interval of t.
(a) Let
u(t) = r(t) sin θ(t), u�(t) = r(t)cos θ(t).
Show that
dθ/dt = cos2 θ + p(t) cos θ sin θ + q(t) sin2 θ,
(1/r)dr/dt = −p(t) cos2 θ + (1 − q(t)) cos θ sin θ.
(b) Using part (a) discuss that if q(t) > p2(t)/4 then solutions are oscillatory and if q(t) < 0 then
solutions are nonoscillatory.
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