Math 5553, Homework 3, Due on 3/25/2014
1. (4 points) Let a, b, c, d, e, f be real numbers and a > 0, d > 0, f > 0. Dene a symmetric
matrix A as follows:
2
a 0 0
a b c
a
ab
ac
bc + de
A = b d 0 0 d e = ab b2 + d2
2 + e2 + f 2
c e f
0 0 f
ac bc + de c
Ap
Math 5553, Homework 5, Due on 4/24/2014
1. (8 points) Consider the m m tridiagonal
a1 b1
c1 a2
0 c2
T =
0 0
matrix
0
b2
a3
.
.
0
.
.
.
.
0
0
b3
.
.
.
.
0
0
0
0
bm1
am
where ai , bi , ci are real numbers, and bi ci > 0 for all i. This matrix is not nece
Math 5553, Homework 1, Due on 2/4/2013
1. (4 points) Prove that A
n
j=1 |ai,j |
= max1im
and . Prove that for all x Cm ,
x 2 x 1 m x 2
x x 2 m x
1,
x
2. (6 points) Consider vector norms
, for A Cm,n .
2
x
1
m x
3. (4 points) Show that if a matrix is bot
Math 5553, Homework 2, Due on 2/20/2014
1. (2 points) Let A Cmm and B Cnn be two unitary matrices. Show that the (m + n)
A 0
(m + n) matrix
is also unitary.
0 B
2. (4 points) You may use the facts that det(AB) = det(A)det(B) and det(A ) = det(A).
(a) Wha
Math 5553, Homework 4, Due on 4/3/2014
1. (8 points) For the conjugate gradient method, prove that
spancfw_p0 , p1 , . . . , pi = spancfw_r0 , r1 , . . . , ri
You can use all results that we have already proved in class.
2. (12 points) Use the same sett