Math 577 - HW 6 Thanks for

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 normalization rjj, ujj > 0. Is it true or false that R = U? Solution: It is true. Since A is nonsingular, it will have a unique QR factorization with rjj > 0. Then we have A∗A = R∗Q∗QR = R∗R because Q is a unitary matrix. On the other hand, it is easy to see that A∗A is hermitian. And ∀x 6= 0, we have Ax 6= 0 because A is nonsingular. Thus, we have