**1. **

**(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.

**4. **

**(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.

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