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Math 170A, Winter 2020 Sample Questions for Final

The following are a set of questions that are similar to questions that will be on the Final.

Q1. a) Let p be a permutation of 1, 2, . . . , n, and P be the permutation matrix defined

by p. That means that each entry of P is either 0 or 1, with the property that

for every row i, 1 ≤ i ≤ n, P(i, p(i)) = 1 and P(i, j) = 0 for all j 6= p(i).

Show that P

TP = I.

b) Let A be an n × n, symmetric matrix and let B be the result of A after just one

step of Gaussian elimination (so B satisfies bi1 = 0 for all i = 2, 3, . . . , n). Prove

that bij = bji, for all 2 ≤ i, j ≤ n. Be sure to label exactly where you use the

symmetry of A.

Q2. Consider an iterative method for solving Ax = b defined by a “splitting” A = M − N,

and the iterating equation

Mx(k+1) = Nx(k) + b ,Use the

iterating equation Mx(k+1) = Nx(k) + b to obtain a simpler iterating equation for e

(k)

(you will need the facts that Ax∗ = b and A = M − N).

Given M, N above, will this iterative method converge for any initial condition x. Will the Gauss Seidel iteration converge regardless of the

initial guess x

(0)? Explain your answer.

Q4. Write a MATLAB function that takes as input a matrix A, a vector b, an initial guess

x

(0), and a maximal number of iterations k, and then uses the Gauss Seidel method

to solve Ax = b with initial guess x

(0). Your function should stop when the maximal

number k of iterations is reached.

with partial pivoting. Make sure you write out the matrices P, L, U. Remember

that you can always check your answer by computing A = P

TLU.

Q6. To answer the questions below, you may assume that the following holds (you don’t

need to prove this):

kxk∞ ≤ kxk2 ≤ kxk1 ≤√nkxk2 ≤ nkxk∞.

Questions:

1

a) Show that kAk1 ≤ nkAk∞.

b) Find an example of a matrix for n = 2 such that kAk1 = 2kAk∞. You must

compute both norms to show that your example works.

Hint: It may help to first find an example of a length 2 vector x for which

kxk1 = 2kxk∞ first.

c) Let Q be an orthogonal matrix. Show that kQxk2 = kxk2 for every vector x.

Q7. Let u be a unit-norm vector (||u||2 = 1) and let Qu be the Householder reflector

Qu = I − 2uuT

. What are the eigenvalues and determinant of Qu? (Hint: recall that

P = uuT

is a rank-one matrix for which P u = u.)

Q8. Consider the positive definite matrix A ∈ R

n×n

, and let A = RTR be its Cholesky

decomposition.

a) Show that κ2(A) = (κ2(R))2

, where κ2 stands for “condition number in norm 2”.

b) Show that the right singular vectors of A are also right singular vectors for R.

Q9. Suppose that the matrix A ∈ C

4×4 has eigenvalues {2, −2, 1.5, 0.5}, with eigenvectors

v1, v2, v3, respectively, v4. Let q = v3 + 2v4. Will the power method started with q0 = q

converge, and if so, to what? (Hint: without normalizing the vectors, try writing out

q1, q2, q3, q4, . . .)

a) Perform the Gram-Schmidt process on {v1, v2, v3} to obtain three orthonormal

vectors q1, q2, q3.

b) Let A = [v1, v2, v3]; use a) to find the minimizer x for the least squares problem

involving A and b = [1, 1, 1, 1]T

.

Q11. The following is an excerpt from a MATLAB code:

[U, S, V] = svd(A);

S = diag(S);

r = rank(A);

S1 = S(1:r,1:r);

S2 = diag(ones(r,1)./S(1:r));

X = V(:,1:r)*S2*U(:,1:r)’;

Y = U(:,1:r)*S1*V(:,1:r)’;

Read and understand the code, then answer the following questions:

2a) What is X?

b) In exact arithmetic, what is norm(A-Y)?

Q12. Let A ∈ C

2×2 be a defective matrix. Show that A is similar to a matrix,

for some particular choices of λ and α 6= 0 in C

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