Math 477 — Homework Assignment 4, due Oct.26, 2006
1. Given A∈Cm×nof rank nand b∈Cm, consider the block 2 ×2 system of equations
I A
A∗O r
x=b
0,
where Iis the m×midentity matrix. Show that this system has a unique solution [r,x]T, and
that the vectors rand xare the residual and the solution of the least squares problem:
Given A∈Cm×nof full rank, m≥n,b∈Cm, find x∈Cnsuch that kb−Axkis
minimized.
2. Here is a stripped-down version of one of MATLAB’s built-in m-files.
[U,S,V] = svd(A);
S = diag(S);
tol = max(size(A))*S(1)*eps;
r = sum(S > tol);
S = diag(ones(r,1)./S(1:r));
X = V(:,1:r)*S*U(:,1:r)’;
Explain line-by-line what this code does. What is X?
3. Suppose an m×mmatrix Ais written in the block form A=A11 A12
A21 A22 , where A11 is n×n
and A22 is (m−n)×(m−n). Assume that Ais such that its LU factorization exists. Verify
the formula I O
−A21A−1
11 I A11 A12
A21 A22 =A11 A12
O A22 −A21A−1
11 A12
for “elimination” of the block A21. The matrix A22 −A21A−1
11 A12 is known as the Schur comple-
ment of A11 in A.
4. Let Abe the 4 ×4 matrix
A=
−1 1 0 −3
1 0 3 1
0 1 −1−1
3 0 1 2
.
(a) Compute the LU factorization of Awith and without partial pivoting.
(b) Determine det(A) from the 2 LU factorizations of Aobtained in (a).
(c) Describe how Gaussian elimination with partial pivoting can be used to find the determinant
of a general square matrix.
5. Given a nonsingular matrix A. Describe how to find the inverse of Afrom its LU factorization
A=LU without explicitly computing inverse matrices.
6. Let Abe a nonsingular square matrix and let A=QR and A∗A=U∗Ube QR and Cholesky
factorizations, respectively, with the usual normalizations rjj, uj j >0. Is it true or false that
R=U? Explain.
7. Give an example of a symmetric positive matrix that is not positive definite, i.e., construct a
(2 ×2) matrix Awith all positive entries such that xTAxis sometimes negative.