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

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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
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(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 .
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