For each of the following vectors x, find a rotation matrix R such that Rx = _x_2e1: (a)

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 ri j = _ x2 i + x2j _1/2 i, j = 1, 2, 3 For each of the following, determine a Givens transformation Gi j such that the ith and j th coordinates of Gi jx are ri j 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 such that Hx = e1, where = _x_2: (a) x = (8,1,4)T (b) x = (6, 2, 3)T (c) x = (7, 4,4)T

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

Let A = 3 3 2 1 1 1 1 5 1 5 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 2 4 2 6 7 2 1 8 and b = 9 9 3 (a) Use Householder transformations to transform A into an upper triangular matrix R. Also transform the vector b; that is, compute b(1) = H2H1b. (b) Solve Rx = b(1) for x and check your answer by computing the residual 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 n 2 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 n n matrix, how many operations are required to compute GA and HA?

Let Hk = I 2uuT 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) Hk A? 1

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) QTA? 1

Let R1 and R2 be two 2 2 rotation matrices and let G1 and G2 be two 22 Givens transformations. What type of transformations are each of the following? (a) R1R2 (b) G1G2 (c) R1G1 (d) G1R1 1

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_22 = 2(x y)T x (b) Qx = y 1

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)? 1

Let R be an n n plane rotation. What is the value of det(R)? Show that R is not an elementary orthogonal matrix. 1

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

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 H1_H2, where H1 and H2 are Householder transformations and 1 = _x_ _y_, 2 = 3 = = n = 0 1

In constructing a Householder matrix to zero out the last n 1 of a vector x in Rn, the scalar must satisfy || = _x_2. Instead of always taking = _x_2, it is customary to choose = _x_2 whenever x1 > 0. Explain the advantages of choosing in this manner. 1

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.