Math 782, HW 2, Due Wednesday, Feb. 21
For problems involving MATLAB, turn in printouts of your m-files and obtained numerical
results.
Problem 1. Find, by hand, a SVD of A="1 1
1 1 #.
Problem 2. Two matrices A, B ∈Rm×mare said to be orthogonally similar if A=QBQT
for some orthogonal Q∈Rm×m.
(a) Show that if A, B ∈Rm×mare orthogonally similar, then they have the same singular
values.
(b) Show by an example that the converse of (a) is not true. Hint: Use A, B ∈R1×1.
Problem 3. Assume that A∈Rm×mis nonsingular. Show that all singular values σjof A
are positive and that kA−1k2= 1/σm.
Problem 4. Use an appropriate built-in function of MATLAB to compute a reduced (econ-
omy size) SVD of
A=
1 2 3 4
5 6 7 8
9 10 11 12
1 1 1 1
3 2 1 0
.
Using this SVD determine: Rank of A,kAk2, orthonormal bases for Range of Aand Null of
A.
Problem 5. Assume A∈Rm×n,m≥n, and let A=ˆ
Qˆ
Rbe a reduced QR factorization of
A. Show that Rank of A=nif and only if all the diagonal entries of ˆ
Rare nonzero.
Problem 6. Assume A∈Rm×n,m≥n, Rank of A=n, and let A=ˆ
Qˆ
Rbe a reduced QR
factorization of A. Show that the columns of ˆ
Qform an orthonormal basis for the Range of
A.
Problem 7. Write two MATLAB functions [Q, R] =clgs(A) and [Q, R] =mgs(A) that, for a
given A∈Rm×nwith m≥nand Rank A=n, compute reduced QR factorizations A=ˆ
Qˆ
R
using classical and modified Gram-Schmidt processes, respectively (cf. Algorithms 7.1 and
8.1 in the text). Test your functions on the matrix
A=
111
0 0
00
0 0
, = 10−4,10−6,10−8,
by comparing ˆ
Qˆ
Rto A. Assess “orthonormality” of the columns of ˆ
Qby computing ˆ
QTˆ
Q.
Draw conclusions about which of the two Gram-Schmidt processes is numerically better.
(Use the long format of MATLAB to display your results.)
