Example 5.11.2 Problem: Explain how to make C lower triangular in (5.11.11). Solution: Apply Householder (or Givens) reduction to produce an orthogonal matrix Pmxm such that PA = (B), where B is rxn of rank r. House- holder (or Givens) reduction applied to BT results in an orthogonal matrix Qrxn and a nonsingular upper-triangular matrix T such that B Q? -TXT 0 TT 0 0 - (Tox") = B = (T"|0) (6) -("7" :) so A=p () = P1(1.*. :) = P1 (1 :)Q is a URV factorization. B T? 0 Note: C can in fact be made diagonal see (p. 412). 5.11.8. Use the Householder reduction method as described in Example 5.11.2 to compute a URV factorization as well as orthonormal bases for the -4-2-4-2 four fundamental subspaces of A = 2 -2 2 1 1 -4 -2 Example 5.11.2 Problem: Explain how to make C lower triangular in (5.11.11). Solution: Apply Householder (or Givens) reduction to produce an orthogonal matrix Pmxm such that PA = (B), where B is rxn of rank r. House- holder (or Givens) reduction applied to BT results in an orthogonal matrix Qrxn and a nonsingular upper-triangular matrix T such that B Q? -TXT 0 TT 0 0 - (Tox") = B = (T"|0) (6) -("7" :) so A=p () = P1(1.*. :) = P1 (1 :)Q is a URV factorization. B T? 0 Note: C can in fact be made diagonal see (p. 412). 5.11.8. Use the Householder reduction method as described in Example 5.11.2 to compute a URV factorization as well as orthonormal bases for the -4-2-4-2 four fundamental subspaces of A = 2 -2 2 1 1 -4 -2