Use the classical or modi ed Gram-Schmidt process to compute a reduced QR factorization of A

SKU: assim5 Category:

Use the factorization PA = LU obtained in (a) to solve the system Ax = b

 

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1. a-let

 

(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

32

 

(a) Compute B = ata and compute the upper triangular H for the Cholesky factorization B =hth

 

(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 =at

 

(c) The matrix A has a singular value decomposition (SVD) A = U∑V T ; where,

uv

Use the SVD to compute PA and PN.

 

4. ab1

 

(a) Use the classical or modi ed 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 bxAlso 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 satis es QT1Q1 = In and R1  rn is upper triangular. Prove cae Is this a Cholesky factorization?

(b) Let A = 011. The eigenvalues of A are 1 and 1. Let E = 0e. Determine the two eigenvalues λ1 and λ2 of A + E =

1e. 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

a0

(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) p3 = 0 for j = 1; 2; 3.) Prove

1j

Hint: Apply Gerschgorin’s 1st theorem to A.

 

 
 

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