(a) Use the Gaussian elimination with partial pivoting to determine a permutation P (in explicit matrix form), a unit lower triangular matrix L, and an upper triangular matrix U such that PA = LU. You need to provide P, L, and U.
(b) Use the factorization PA = LU obtained in (a) to solve the system Ax = b. You need to provide the numerical solution x.
2.Let A and b be given as in Problem 1.
(a) Use the Gaussian elimination with complete pivoting to determine permutation matrices P and Q (in explicit matrix forms), a unit lower triangular matrix L, and an upper triangular matrix U such that PAQ = LU. You need to provide P, Q, L, and U.
(b) Use the factorization PAQ = LU obtained in (a) to solve the system Ax = b. You need to provide the numerical solution x.
3.(You need to provide exact answers for this problem!) Let
(a) Compute B = and compute the upper triangular H for the Cholesky factorization B =
(b) Compute PA, the orthogonal projection onto the range of A, and PN, the or thogonal projection onto to the nullspace of AT , by using the formula PA =
(c) The matrix A has a singular value decomposition (SVD) A = U∑V T ; where,
Use the SVD to compute PA and PN.
(a) Use the classical or modied Gram-Schmidt process to compute a reduced QR factorization of A.
(b) Use the Householder matrices or Givens rotations to compute a reduced QR factorization of A.
(c) Use the reduced QR factorization obtained in either (a) or (b) to solve the least squares problem Also compute the optimal residual r = b – Ax.
5.(a) Consider A 2 Rmn with m n and rank(A) = n. The matrix C = (ATA)-¹ arises in many statistical applications known as the variance-covariance matrix.Suppose A = Q1R1 is a reduced QR factorization, where Q1 satises QT1Q1 = In and R1 is upper triangular. Prove Is this a Cholesky factorization?
(b) Let A = . The eigenvalues of A are 1 and 1. Let E = . Determine the two eigenvalues λ1 and λ2 of A + E =
. Then find |λ1 – 1| and |λ2 – 1|.
6.(Bonus problem, 10 points) Let p(λ) = λ³+ a2λ² + a1λ + a0 be a monic degree 3 polynomial of λ , and let
(a) Prove det (λI3 -A) = p(λ).(This shows that all the zeros of p(λ) are just the eigenvalues of A.)
(b) Suppose λ1 , λ2 ,λ3 are the zeros of p(λ) = 0 for j = 1; 2; 3.) Prove
Hint: Apply Gerschgorin’s 1st theorem to A.