(a) In MATLAB, the simplest way to compute a Householder matrix that zeroes out entries

Chapter 7, Problem 10

(choose chapter or problem)

(a) In MATLAB, the simplest way to compute a Householder matrix that zeroes out entries of a given vector x is to compute the QR factorization of x. Thus, if we are given a vector x Rn, then the MATLAB command [H, R] = qr(x) will compute the desired Householder matrix H. Compute a Householder matrix H that zeroes out the last three entries of e = ones(4, 1). Set C = [e, rand(4, 3)] Compute H e and H C. (b) We can also compute the vector v and the scalar b that determine the Householder transformation that zeroes out entries of a given vector. To do this for a given vector x, we would set a = norm(x); v = x; v(1) = v(1) a b = a (a x(1)) Construct v and b in this way for the vector e from part (a). If K = I 1 b vvT , then Ke = e _ vT e b _ V Compute both of these quantities with MATLAB and verify that they are equal. How does Ke compare to He from part (a)? Compute also K C and C v ((v_ C)/b) and verify that the two are equal. 1