ColoState
Spring 2016
Math 561
Homework 1
Due Mon. 02/01/2016
Name:
CSUID:
Problem 1 (20 points) Assume that x ∈ Rn . Prove the following
√
(i) ∥x∥2 ≤ ∥x∥1 ≤ n∥x∥2 .
√
(ii) ∥x∥∞ ≤ ∥x∥2 ≤ n∥x∥∞ .
Problem 2 (15 points)
Let A = [aij ] ∈ R
m×n
. Prove that ∥A∥1 = max
1≤j≤n
m
∑
|aij |.
i=1
Problem 3 (15 points) Let A be an orthogonal matrix. Show that det(A) = ±1. Show
that if B is also an orthogonal matrix and det(A) = −det(B), then A + B is singular.
Problem 4 (15 points) A matrix is strictly upper triangular if it is upper triangular with
zero diagonal entries. Prove that if A is an n × n strictly upper triangular matrix, then
An = 0.
Problem 5 (20 points) List three storage formats for sparse matrices and compare their
advantages and disadvantages (or main features) in storage and matrix-vector multiplications. Use a small size matrix to illustrate these storage formats. You could use an Internet
search engine to find what is available/popular, but this short essay must be written in your
own words.
Problem 6 (15 points)
Let A ∈ Rn×n be defined

2 −1
0
0
 −1
2
−1
0

 0 −1
2 −1
A=
 ··· ··· ··· ···

 0
0
0
0
0
0
0
0
as follows
···
0
0
···
0
0
···
0
0
··· ··· ···
···
2 −1
· · · −1
2




.



Let θ = π/(n + 1). Prove that λk = 2(1 − cos(kθ)), k = 1, 2, . . . , n are the eigenvalues of A
with corresponding eigenvectors
zk = [sin(jkθ)]nj=1 ,
k = 1, 2, . . . , n.