Homework problems that will be graded (Q1 - Q5, 30pts in total):

Q1. Let A ∈ Rn×m , n ≥ m, rank(A) = m. Write out and explain the SVD of the matrix (AT A)−1AT  in terms of the SVD of A = UΣVT .

a) Based on the definitions in class, conclude that the pseudoinverse of A, A†   = (AT A)−1AT .

b) Use (a) to calculate A† A.

c) Use the SVDs of A and A†  to calculate AA† , simplifying as much as possible.

Q2.  Consider the matrix

A =  l  」

「  1     1     .

Compute ∥A∥2  and κ2 (A).

Q3. Let v,w  ∈ Rn   and define A  = vwT .   Prove that  ∥A∥2   =  ∥v∥2 ∥w∥2 .   Is the same

result true for the Frobenius norm, i.e. is  ∥A∥F   =  ∥v∥2 ∥w∥2 ?  Prove this or give a counterexample.

Hint: For the first part, figure out the rank and the SVD of A. For the second part: consider the result from HW5, Q2 a.

Q4.  (This is similar to exercise 5.1.24 from the textbook, and it is meant to show you why

computing roots of polynomials is numerically unstable.)

Let A = [0(1)   1(0) ], the 2 × 2 identity matrix.

(a) Show that the characteristic equation of A is

λ2 − 2λ + 1 = 0.

It has a single, repeated, eigenvalue: λ 1  = λ2  = 1.

(b) We now perturb one coefficient of the characteristic polynomial slightly and con- sider the equation

λ2 − 2λ + (1 − ε) = 0,

where 0 < ε ≪ 1. Solve the equation for the roots 1  and 2 , in terms of ε .

(c) Show that when ε = 10 −12 , | 1  − λ1 | and | 2  − λ2 | are one million times bigger than ε .

(d) Sketch the graphs of the original and perturbed polynomials (using some ε bigger than 10 −12, for example ε = 0.01), to give some indication why the roots are so sensitive to the ε pertubation.

You can use the ”plot”feature in MATLAB, or any other software you wish.

Q5.  (MATLAB problem)

Let

l 2(1)   2

A =

「 0   1

4 」

2(2)   .