## Description

1. Here we continue the problem studied in Activity 11. Let a 4-by-2 matrix X have

SVD X = USV T where U =

1

2

1 1

1 −1

1 −1

1 1

, S =

”

1 0

0 γ

#

, and V = √

1

2

”

1 1

1 −1

#

Let y =

1

0

0

1

.

a) The ratio of the largest to the smallest singular values is termed the condition

number of X. Find the condition number if γ = 0.1, and γ = 10−8

. Solve

Xw = y for w and find ||w||2

2

for these two values of γ.

b) A system of linear equations with a large condition number is said to be “illconditioned”. One consequence of an ill-conditioned system of equations is solutions with large norms as you found in the previous part of this problem.

A second

consequence is that the solution is very sensitive to small errors in y such as may

result from measurement error or numerical error. Suppose y =

1 +

0

0

1

.

Write

w = wo + w where wo is the solution for arbitrary γ when = 0 and w

is the

perturbation in that solution due to some error 6= 0. How does the norm of the

perturbation due to 6= 0, ||w

||2

2

, depend on the condition number? Find ||w

||2

2

for = 0.01 and γ = 0.1 and γ = 10−8

.

c) Now consider a “low-rank” inverse. Instead of writing

(XTX)

−1XT =

X

p

i=1

1

σi

viu

T

i

where p is the number of columns of X (assumed less than the number of rows),

we approximate

(XTX)

−1XT ≈

Xr

i=1

1

σi

viu

T

i

In this approximation we only invert the largest r singular values, and ignore all

of them smaller than σr. Use r = 1 in the low-rank inverse to find w = wo + w

where y =

1 +

0

0

1

as in part b). Compare the results to part b).