Homework #5. Due February 12

Problems u.* are from the Review of Chapter u, prblems u.x.* are from section x in chapter u.

Part A: 3.14, 3.18, 3.28, 3.39, 4.2.1, 4.2.10

Part B: Strang 3.4.5, 3.4.8, 3.4.14, 3.4.18, 3.4.21, 3.4.22, 4.2.6, 4.2.14
and the following problem:
9. Cholesky factorization.
Suppose A is invertible.
a. Show that a symmetric (real) matrix A can be represented (factorized) into
A= L D LT.
b. Suppose the diagonal matrix D has nonnegative values on the diagonal. Then we can define D1/2. How, and in how many ways?
c. Let S=L D1/2, show that A= S ST. This is the Cholesky factorization: a symmetric "positive definite" matrix A is a product of two triangular factors S and its transpose.
d. Show (using Gram-Schmidt) that if B = AT A , then the Cholesky factorization (up to the choice of D1/2) for B is B =S ST where S= RT, and R is exactly the matrix in the QR factorization of A: A= QR.