Description
1. Can three vectors in the xy plane have u · v < 0, v · w < 0 and u · w < 0?
2. Let c ∈ R. Suppose that A is an n × n matrix and that the sum of the entries in each
column of A is c. Prove that c is an eigenvalue of A.
Hint: Consider the sum of the row vectors of the matrix A − cI.
3. If A(t) is a continuously-differentiable n × n matrix function that is invertible at each
1
t, show that
d
dtA
−1
(t) = −A
−1
(t) A˙(t) A
−1
(t)
4. If λ is an eigenvalue of A and X is the corresponding eigenvector, then prove that λ−s
is an eigenvalue of A − sI for any scalar s and X is the corresponding eigenvector.
5. For any two n × n matrices, say A and B:
(a) are real-symmetric matrices, both AB and BA always have the same eigenvalues.
True or False?
(b) matrix B is invertible, AB and BA always have the same eigenvalues. True or
False?
(c) matrix B is invertible, AB and BA always have the same eigenvectors. True or
False?
Give support for all your answers.
6. If rows of an m × n matrix A are linearly independent,
(a) Is Ax = b necessarily solvable?
(b) If Ax = b is solvable, is the solution necessarily unique?
7. Show that if A is a non-singular matrix, and λ is an eigenvalue of A, then 1
λ
is an
eigenvalue of A−1
.
8. Create an 8×8 matrix H using the command hilb(8) in Matlab. Generate a random
vector x, and compute Hx = b. Add a tiny amount of noise to b. Then recover x from
b by running the command ˆx = H−1
b. How accurate is the recovered x? Why did this
happen? You don’t need to provide any code or console output, just describe what you
did and what you got in a few sentences.
9. Suppose we want to recover the solution to the system Ax = b. We don’t know b
exactly, but we have a noisy measurement vector ˆb. To do this, we could compute
xˆ = A−1
b. Prove that
kx − xˆk
kxk
≤ κ
kb − ˆbk
kbk
Here κ is the condition number of A.
Hint: Definition of condition number.
10. Consider the measurement model
y = Dx + η
where D ∈ R
m×n
is a measurement matrix, and η ∈ R
n×1
is a noise vector with
distribution
η ∼
1
p
(2π)m|Σ|
e
− 1
2
η
T Σ−1η
2
(a) (b)
Figure 1
for some covariance matrix Σ, where |Σ| denotes the determinant of Σ. Suppose we
have prior knowledge that each entry in x is draw from an I.I.D. Laplace distribution
xi ∼
1
2b
e
−
|x|
b
Derive the Negative Log-Likelihood (NLL) function for x given y. Write the complete
NLL without throwing away any constants (although you may use Bayes rule, which
implicitly throws away a normalization constant).
11. Prove that the shortest path between two points on a the surface of the sphere is
the straight line (curve) on the sphere surface (in spherical co-ordinates), using the
Euler-Lagrange equation.
12. Both the plots in Figs. 1(a) and 1(b) are derived from f(x) = 1
2
x
T Hx + g
T x + c
(a) What are the constraints on H for (a) and for (b)?
(b) Find the optimal value in (a) and (b), if they exist.
13. Suppose the random column vectors X, Y live in R
n and R
m respectively, and the
vector (X, Y ) in R
n+m has a multivariate normal distribution whose covariance is the
symmetric positive-definite matrix
Σ = ”
A B
BT C
#
3
Here A ∈ R
n×n
is the covariance matrix of X, C ∈ R
m×m is the covariance matrix of Y
and B ∈ R
n×m is the covariance matrix between X and Y . Prove that the conditional
covariance of X given Y is the Schur complement of C in Σ. Hint: Definitions of Schur
complement, conditional covariance.
2 Submission Guidelines
Submit your assignment named according the format YourDirectoryName hw0.zip onto
ELMS/Canvas (Please compress it as .zip compressed file format. Other formats
will not be acceptable.) Your DirectoryName is the username to your UMD e-mail ID. If
your email ID is johnsmith@terpmail.umd.edu, your DirectoryName is johnsmith. Your
zip file should contain PDF document typeset in LATEX. If your submission does not comply
with the guidelines, you’ll be given ZERO credit.
3 Collaboration Policy
You can discuss with any number of people. But the solution you turn in MUST be your
own. Plagiarism is strictly prohibited. Plagiarism checker will be used to check your submission. Please make sure to cite any references from papers, websites, or any other student’s
work you might have referred.
4