CSci 5304, F12 Second Midterm Test Nov. 14th, 2012

Class lecture notes (those posted on the class web site) are allowed. No books. Duration: 75mn. Calculators not needed. Weights are indicated in brackets at the end of each question.Base = 100pts.

1. Short questions. [Answer True or False or give a very brief answer. No justificationsneeded.] [24 pts] (4pts per question)

a. The singular values of A do not change if one or several columns of A change signs(i.e., if these columns are multiplied by -1).

b. The Householder QR factorization of an m × n matrix, with m < n, does not alwaysexit.

c. If A is symmetric positive definite (SPD) and α is a positive scalar then A+αI is SPD.

d. If A is a symmetric positive definite matrix then trace(A) > 0.

e. If Q is an m× n orthogonal matrix with m > n then ‖QTx‖2 = ‖x‖2 for any x ∈ Rm.

f. Let A be an m× n matrix and U be an m×m unitary matrix. Then UA and A havethe same singular values.

2. Consider the matrix A(α, β) shown on the right.We seek α, β so that ‖I − A(α, β)‖F is minimum,where I is the 3× 3 identity matrix.(a) Express the problem as a least-squares system;[12 pts] Then (b) find its solution.[8 pts]

A(α, β) =

α− β 0 α0 1 0β 0 α + β

3. Let x = [−1 0 − 2 2]T . In the following w is always a unit vector (‖w‖2 = 1).(a) Find a unit vector w and α > 0 such that (I − 2wwT )x = αe1. [8 pts](b) Find a unit vector w and α < 0 such that (I − 2wwT )x = αe2 [8 pts](c) Find a unit vector w such (I − wwT )x = 0. [8 pts]

4. Consider the matrix A =

[2 −1 2 12 1 2 −1

](a) Compute AAT and obtain the matrix U of the SVD of A. [6 pts]

(b) Find the “thin” SVD of A: A = UΣ1VT1 , U ∈ R2×2 unitary, Σ1 ∈ R2×2,... [6 pts]

(c) What is the pseudo-inverse of A? [6 pts]

(d) Find the vector xLS which has the smallest 2-norm and for which ‖b−Ax‖2 is minimumfor the right-hand side b = [8 8]T . [7 pts](e) Find *all* least-squares solutions x to min ‖b − Ax‖2, for the same b as in (d). [Write

these solutions in terms of free parameters.] [7 pts]