Exercise 20.2
We can say that L is lower triangular and also lij = 0 for i−j > p. We can say that U is upper triangular and also that uij = 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 aij = Xlikukj = 0 for i = 1,...,m − p − 1 .
k=1
Therefore, lij = 0 for i−j > p. Because L is lower triangular, when j−i > p we have
i aij = Xlikukj = 0, for i = 1,...,m − p − 1 .
k=1
Therefore, uij = 0 for j − i > p.
Exercise 23.1
Let A be a nonsingular square matrix and let A = QR and A∗A = U∗U be
QR and Cholesky factorizations, respectively, with the usual normalizations rjj,ujj > 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 A∗A = (QR)∗QR =
R∗R. Also, because A is nonsingular, Ax 6= 0 for every x 6= 0. Then
0, which shows that A∗A is positive definite. And
because (A∗A)∗ = A∗(A∗)∗ = A∗A, A∗A is also hermitian. So by Theorem 23.1, the Cholesky factorization A∗A = U∗U of A∗A is unique. But then
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A∗A = R∗R = U∗U and both factorizations must be unique, we conclude that R = U.
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