ColoState
Spring 2016
Math 561
Homework 6
Due Mon. 05/02/2016
Name:
CSUID:
(15 points) Problem 1 Describe how to use SVD to find the Moore-Penrose pseudo-inverse.
Give an example of a small-size matrix.
(15 points) Problem 2 Let A be an upper Hessenberg matrix. We apply QR-factorization
to A based on Householder reflections to get Q and R. Prove that Q is also an upper
Hessenberg matrix.
(15 points) Problem 3
(Corrected Version) Suppose A is an n × n upper Hessenberg
matrix and we carry out QR-factorization based on Householder reflections. Show that the
operations count for this QR-factorization (for upper Hessenberg matrix) is 3n2 + O(n).
(20 points) Problem 4 Test the Francis QR-iteration method and the power method on
the following 3 × 3 matrix


0 0 1
A =  1 0 0 .
0 1 0
Report your findings on the numerical performance of these methods. What techniques for
improvement (e.g., shifting) do you suggest for this type of matrices?
(15 points) Problem 5 Let x, y be n-dimensional real column vectors and I be the n × n
identity matrix. Prove the following formula about their inner and outer products:
det(I + xy T ) = 1 + y T x.
[
]
D uT
(20 points) Problem 6
Let A =
, where D = diag(d1 , . . . , dn ) has distinct
u dn+1
diagonal entries and u ∈ Rn has nonzero components. Show that
(i) If α is an eigenvalue of A, then det(D − αI) ̸= 0.
(ii) If α is an eigenvalue of A, then α is a root of the following equation
f (λ) = λ − dn+1 +
n
∑
i=1
u2i
= 0.
di − λ
Remark: Team work (two people on a team) is allowed for Prob.#3,5,6.