Solved: For each of the following systems, use a Givens reflection to transform the

For each of the following vectors x, find a rotation matrix R such that Rx = x 2e1: (a) x = (1, 1)T (b) x = ( 3, 1)T (c) x = (4, 3)T

Given x R3, define rij =  x2 i + x2 j 1/2 i, j = 1, 2, 3 For each of the following, determine a Givens transformation Gij such that the ith and jth coordinates of Gijx are rij and 0, respectively: (a) x = (3, 1, 4)T , i = 1, j = 3 (b) x = (1, 1, 2)T , i = 1, j = 2 (c) x = (4, 1, 3)T , i = 2, j = 3 (d) x = (4, 1, 3)T , i = 3, j = 2

For each of the given vectors x, find a Householder transformation that zeros out the last two entries of the vector. (a) x = (1, 8, 4)T (b) x = (3, 6, 2)T (c) x = (0, 3, 4)T

For each of the following, find a Householder transformation that zeroes out the last two coordinates of the vector: (a) x = (5, 1, 4, 8)T (b) x = (4, 3, 2, 1, 2)T

Let A = 1 3 2 111 1 5 1 1 1 2 (a) Determine the scalar and vector v for the Householder matrix H = I (1/)vvT that zeroes out the last three entries of a1. (b) Without explicitly forming the matrix H, compute the product HA.

Let A = 1 3 2 1 2 28 8 2 7 1 and b = 11 2 0 1 (a) Use Householder transformations to transform A into an upper triangular matrix R. Also, transform the vector b; that is, compute c = H2H1b. (b) Solve Rx = c for x and check your answer by computing the residual r = b Ax.

For each of the following systems, use a Givens reflection to transform the system to upper triangular form and then solve the upper triangular system: (a) 3x1 + 8x2 = 5 4x1 x2 = 5 (b) x1 + 4x2 = 5 x1 + 2x2 = 1 (c) 4x1 4x2 + x3 = 2 x2 + 3x3 = 2 3x1 + 3x2 2x3 = 1

Suppose that you wish to eliminate the last coordinate of a vector x and leave the first n2 coordinates unchanged. How many operations are necessary if this is to be done by a Givens transformation G? A Householder transformation H? If A is an nn matrix, how many operations are required to compute GA and HA?

Let Hk = I2uuT be a Householder transformation with u = (0, ... , 0, uk, uk+1, ... , un) T Let b Rn and let A be an n n matrix. How many additions and multiplications are necessary to compute (a) Hkb?; (b) HkA?

Let QT = Gnk G2G1, where each Gi is a Givens transformation. Let b Rn and let A be an n n matrix. How many additions and multiplications are necessary to compute (a) QT b; (b) QT A?

Let R1 and R2 be two 2 2 rotation matrices and let G1 and G2 be two 2 2 Givens transformations. What type of transformations are each of the following? (a) R1R2 (b) G1G2 (c) R1G1 (d) G1R1

Let x and y be distinct vectors in Rn with x 2 = y 2. Define u = 1 x y 2 (x y) and Q = I 2uuT Show that (a) x y 2 2 = 2(x y) T x (b) Qx = y

Let u be a unit vector in Rn and let Q = I 2uuT (a) Show that u is an eigenvector of Q. What is the corresponding eigenvalue? (b) Let z be a nonzero vector in Rn that is orthogonal to u. Show that z is an eigenvector of Q belonging to the eigenvalue = 1. (c) Show that the eigenvalue = 1 must have multiplicity n1. What is the value of det(Q)?

. Let R be an n n plane rotation. What is the value of det(R)? Show that R is not an elementary orthogonal matrix

Let A = Q1R1 = Q2R2, where Q1 and Q2 are orthogonal and R1 and R2 are both upper triangular and nonsingular. (a) Show that QT 1Q2 is diagonal. (b) How do R1 and R2 compare? Explain.

Let A = xyT , where x Rm, y Rn, and both x and y are nonzero vectors. Show that A has a singular value decomposition of the form H1H2, where H1 and H2 are Householder transformations and 1 = x y , 2 = 3 == n = 0

Let R = cos sin sin cos Show that if is not an integer multiple of , then R can be factored into a product R = ULU, where U = 1 cos 1 sin 0 1 and L = 1 0 sin 1 This type of factorization of a rotation matrix arises in applications involving wavelets and filter bases.