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Department of Mathematics Fall 2007
MATH 6370, Section 10583 MW 5:30-7:00PM
A. Caboussat
Homework Chapter 4
Homework from the text
Chapter Exercises
Additional Homework
Problem 1 5
Show that, if a matrix A ∈ C is both triangular and unitary, then it is diagonal.
Problem 2 6
A Hadamard matrix is a matrix whose entries are all ±1 and whose transpose is equal to its inverse time a
constant factor.
It is known that, if A is a Hadamard matrix of dimension m > 2, the m is a multiple of 4. It is not known,
however, whether there is a Hadamard matrix for every such m, though examples have been found for all cases
m 6 424.
Show that the following recursive description provides a Hadamard matrix of each dimension m = 2 k , k =
0 , 1 , 2 ,.. ..
H 0 = (1) , Hk+1 =
Hk Hk
Hk −Hk
Problem 3 8
Consider Gaussian elimination carried out with pivoting by columns instead of rows, leading to a factorization
AQ = LU (instead of P A = LU ), where Q is a permutation matrix.
3.a) Show that if A is nonsingular, such a factorization always exists.
3.b) Show that if A is singular, such a factorization does not always exist.
Problem 4 1
Let a > 0 and c be two real numbers such that a > 2 |c|. Let A be the N × N matrix given by
a c
c a c
c
. (^) c
c a c
c a
and ~b = (b 1 ,... , bN ) T be a given N -vector.
4.a) Show that A is symmetric positive definite and conclude that the linear system A~x = ~b admits a unique
solution. Conclude that the Gauss elimination algorithm can be used.
4.b) Write a Gauss elimination algorithm to solve A~x =
b without using any N
2 array of data. Write a small
program (in Matlab for instance) to illustrate your algorithm on an example.
Problem 5 7
Suppose a m × m matrix A is written in the block form
A =
A 11 A 12
A 21 A 22
where A 11 is n × n and A 22 is (m − n) × (m − n). If A is nonsingular.
5.a) Show the formula
I 0
−A 21 A
− 1 11 I
A 11 A 12
A 21 A 22
A 11 A 12
0 A 22 − A 21 A
− 1 11 A^12
The matrix A 22 − A 21 A
− 1 11 A 12 is known as the Schur complement of A 11 in A.
5.b) Suppose A 21 is eliminated row by row by means of n steps of Gaussian elimination. Show that the
bottom-right (m − n) × (m − n) block of the results is again A 22 − A 21 A
− 1 11
A 12.
Problem 6 2
Let ||| · ||| : MN ×N → R be the application defined by
|||A||| = max X^ ~∈RN^ ,~x 6 =~ 0
||A~x||
||~x||
where ||~x||
2
N j= x
2 j is the Euclidean norm in R
N .
6.a) Show that ||| · ||| is an algebra norm on MN ×N.
6.b) Show that ρ(A) 6 |||A|||, where ρ(A) = max 16 j 6 N |λj | and where λj are the (complex) eigenvalues of A
(ρ(A) is called spectral radius).
6.c) Show that, if A is not singular, then
cond(A) >
max 16 j 6 N |λj |
min 16 j 6 N |λj |
Problem 7 3
Let A ∈ MN ×N ,
b ∈ R
N ,
b 6 = 0 and ~x ∈ R
N the solution to A~x =
b.
Let δA and δ
b be perturbations of A and
b respectively. We assume that |||δA|||
∣A−^1
7.a) Show that A + δA is not singular.
7.b) Show that, if (A + δA)(~x + δ~x) = ~b + δ~b, then
||δx||
||~x||
cond(A)
1 − cond(A)
|||δA||| ||A||
||δb|| ∣ ∣ ∣
b
|||δA|||
|||A|||
where cond(A) = |||A|||
∣A−^1
11.b) How does the L 2 -norm of the matrix A behave as m → ∞? Comment on the inequality ρ(A) 6 |||A|||
when m → ∞.
11.c) Wow does your answer to the first two points change if you consider random triangular matrices instead
of full matrices?
