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Lecture #9 Notes - Matrix Theory Math6304 Lecture Notes...

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Matrix Theory, Math6304 Lecture Notes from September 25, 2012 taken by Katie Watkins Last Time (9/20/12) Cayley Hamilton Block diagonalization with triangular blocks “Nearly” Diagonalizability Q: We saw Schur does not give a unique upper triangular form. Is there a “nice” choice? How can we compute it? 2.4 QR Factorization 2.4.1 Theorem (QR Factorization) . Let A M n , there is a pair of a unitary Q and an upper triangular R in M n such that the diagonal entries of R are non-negative and A = QR , and if A is invertible, then R and Q are unique. Proof. First, lets examine the invertible case. Uniqueness: Let A = Q 1 R 1 = Q 2 R 2 with Q 1 , Q 2 M n unitary and R 1 , R 2 upper triangular with non-negative diagonal entries r (1) j,j 0 and r (2) j,j 0 . The entries on the diagonal cannot be zero because then we would have an eigenvalue of 0 and then A would not be invertible. So r (1) j,j > 0 and r (2) j,j > 0 . We transform the identity to: Q 1 = Q 2 R 2 R 1 1 Which implies that M = Q 2 Q 1 = Q 2 ( Q 2 R 2 R 1 1 ) = R 2 R 1 1 We see that M is unitary and upper triangular, which implies that M is normal. Since M is upper triangular and normal, we know that it is diagonal. Since the eigenvalues of R 2 R 1 1 are strictly positive and have magnitude 1, we know that
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