{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture #10 Notes

# Lecture #10 Notes - Matrix Theory Math6304 Lecture Notes...

This preview shows pages 1–3. Sign up to view the full content.

Matrix Theory, Math6304 Lecture Notes from September 27, 2012 taken by Tasadduk Chowdhury Last Time (09/25/12): QR factorization: any matrix A M n has a QR factorization: A = QR , where Q is unitary and R is upper triangular. In addition, we proved that if A is non-singular, Q and R are unique. Cholesky factorization: any B M n satisfying B = A A , A M n has the factorization B = LL where L M n is lower triangular and non-negative on the diagonal. QR algorithm: details and convergence criteria. Warm-up If A = A 0 M n is normal and QR algorithm converges (entry-wise), what can we say about the limiting matrix A ? Convergence implies that there is a unitary Q and a upper-triangular R such that A = Q R = R Q . (1) From above, we see that Q and R commute. Also, A Q = Q R Q = Q A . (2) So Q and A also commute. Moreover, from (1) we get Q A = R = R Q Q = A Q . (3) By taking adjoints, we get A Q = Q A from (2) and Q A = A Q from (3). Thus, R R = Q A A Q = A Q Q A (by commutativity) = A A = A A (by normality of A ) = A Q Q A = Q A A Q (by commutativity) = R R . 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(We claimed above that A is normal. From last lecture’s proposition of QR algorithm we know that A n ’s are all unitarily equivalent. It can be checked that any matrix unitarily equivalent to a normal matrix is also normal: say X and Y are unitarily equivalent, and X is normal. Then, Y Y = ( U XU ) ( U XU ) = U X UU XU = U X XU = U XX U = U XUU X U = Y Y , and thus Y is normal. Hence, since A n ’s are unitarily equivalent to A 0 = A , and A is normal, and thus the limiting matrix A is also normal.) By normality of R and the fact that it is triangular, R = D with D a diagonal matrix and A = D Q = D Q .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}