Math 477 — Homework Assignment 2, due Sept.28, 2006
1. Determine SVDs of the following matrices. Do not use a computer, and do not use the method
for hand calculations discussed in class. Use only basic properties of the SVD and note that the
matrices are either diagonal matrices or rank-1 matrices:
(a) 3 0
0−2,(b) 2 0
0 3 ,(c)
0 2
0 0
0 0
,(d) 1 1
0 0 ,(e) 1 1
1 1 .
2. In the discussion of matrix norms we claimed that the 2-norm of the matrix
A=1 1
0 1
is approximately 1.6180. Using the SVD, work out (the “by-hand” method is from now on
allowed) the exact values of σmin(A) and σmax(A) for this matrix.
3. Find the SVDs of the following matrices:
A=
400
000
007
000
, B =2 1 , C =5
−4.
4. If Pis an orthogonal projector, then I−2Pis unitary. Prove this algebraically, and give a
geometric interpretation.
5. Consider the matrices
A=
1 0
0 1
1 0
, B =
1 2
0 1
1 0
.
Answer the following questions by hand calculation.
(a) What us the orthogonal projector Ponto range(A), and what is the image under Pof the
vector [1,2,3]∗?
(b) Same questions for B.