We can say that L is lower triangular and also l_ij = 0 for ij > p. We can say that U is upper triangular and also that u_ij = 0 for j i > p. To see this, note that since A satisfies the conditions of exercise 20.1, A has an LU factorization A = LU. When i j > p, because U is upper triangular we have

j a_ij = ^Xl_iku_kj = 0 for i = 1,,m p 1 .

k=1

Therefore, l_ij = 0 for ij > p. Because L is lower triangular, when ji > p we have

i a_ij = ^Xl_iku_kj = 0, for i = 1,,m p 1 .

k=1

Therefore, u_ij = 0 for j i > p.

Exercise 23.1

Let A be a nonsingular square matrix and let A = QR and AA = UU be

QR and Cholesky factorizations, respectively, with the usual normalizations r_jj,u_jj > 0. Then it is true that R = U. To see this, first we note that since A is nonsingular, A has a unique QR factorization and AA = (QR)QR =

RR. Also, because A is nonsingular, Ax 6= 0 for every x 6= 0. Then

0, which shows that AA is positive definite. And

because (AA) = A(A) = AA, AA is also hermitian. So by Theorem 23.1, the Cholesky factorization AA = UU of AA is unique. But then

1

AA = RR = UU and both factorizations must be unique, we conclude that R = U.

2