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Problem 1

Mechanical Engineering

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Problem 1. If A = 1 2 4 -1 and we shift to A-7I, what are the eigenvalues and eigenvectors and
how are they related to those of A? Here, B = A-7I = -6 2 - -1 3.
Problem 2. Give an example to show that the eigenvalues can be changed when a multiple of one
row is subtracted from another. Why is a zero eigenvalue not changed by the steps of elimination?
Problem 3. Suppose A has eigenvalues 0, 3, 5 with corresponding independent eigenvectors u;v;w.
(a) Give a basis for the nullspace and a basis for the column space.
(b) Find a particular solution to Ax = v+w. Also, find all solutions to Ax = v+w.
Problem 4. From the unit vector u = 1 6; 1 6; 36; 56, construct the rank-1 projection matrix P = uuT.
(a) Show that Pu = u. Then u is an eigenvector with l = 1.
(b) If v is perpendicular to u show that Pv = zero vector. Then l = 0.
(c) Find three independent eigenvectors of P all with eigenvalue l = 0.
Problem 5. Choose the second row of A = 0 1 ∗ ∗ so that A has eigenvalues 4 and 7.
Problem 6.
(a) If A2 = I, what are the possible eigenvalues of A?
(b) If this A is 2 by 2, and not I or -I, find its trace and determinant.
(c) If the first row is (3;-1), what is the second row?
Problem 7. Suppose the eigenvector matrix S has ST = S-1. Show that A = SLS-1 is symmetric
and has orthogonal eigenvectors.
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Problem 8. Write the most general matrix that has eigenvectors 1 1 and -1 1.
Problem 9. Suppose there is an epidemic in which every month half of those who are well become
sick, and a quarter of those who are sick become dead. Find the steady state for the corresponding
Markov process:
24
dk+1
sk+1
wk+1
35
=
24
1 1
4 0
0 3
4
12
0 0 1
2
35
24
dk
sk
wk
35
:
Problem 10. Write the 3 by 3 transition matrix for a controls course that is taught in two sections,
if every week 14 of those in Section A and 13 of those in Section B drop the course, and 1 6 of each
section transfer to the other section.
Problem 11. Find the limiting values of yk and zk (k ! ¥) if
yk+1 = :8yk +:3zk y0 = 0
zk+1 = :2yk +:7zk z0 = 5:
Problem 12.
(a) When do the the eigenvectors for l = 0 span the nullspace N (A)?
(b) When do all the eigenvectors for l 6= 0 span the column space C (A)?
Problem 13. For -1 1 1 -1, write the general solution to du=dt = Au, and the specific solution
that matches u(0) = (3;1). What is the steady state as t ! ¥?
Problem 14. If P is a projection matrix, show from the infinite series that
eP ≈ I +1:718P:
Problem 15. Decide on the stability or instability of dv=dt = w;dw=dt = v. Is there a solution that
decays?
Problem 16. In most applications the second-order equation looks like Mu00+Ku = 0, with a mass
matrix multiplying the second derivatives. Substitute the pure exponential u = eiwtx and find the
“generalized eigenvalue problem” that must be solved for the frequency w and the vector x.
Problem 17. From this general solution to du=dt = Au, find the matrix A:
u(t) = c1e2t 2 1+c2e5t 1 1:
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Problem 18. A door is opened between rooms that hold v(0) = 30 people and w(0) = 10 people.
The movement between rooms is proportional to the difference v-w:
dv
dt
= w-v and dw
dt
= v-w:
Show that the total v + w is constant (40 people). Find the matrix in du=dt = Au, and find its
eigenvalues and eigenvectors. What are v and w at t = 1?
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