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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 ,the reisapa i ro faun i ta ry Q and an upper triangular R in M n such that the diagonal entries of R are non-negative and A = QR ,andi f 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 .Theen t r ie sonthed iagona lcanno tbeze robecau sethen 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 R 2 R 1 1 = I = Q 2 Q 1 .Thu s , R 2 = R 1
Background image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online