Transcription of The QR Factorization - USM
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The QR Factorization - USM
Jim LambersMAT 610 Summer Session 2009-10 Lecture 9 NotesThese notes correspond to Section in the FactorizationLet be an matrix with full column rank. The factorizationof is a decomposition = , where is an orthogonal matrix and is an upper triangular are three ways to compute this decomposition:1. Using Householder matrices, developed by Alston S. Householder2. Using Givens rotations, also known as Jacobi rotations, used by W. Givens and originallyinvented by Jacobi for use with in solving the symmetric eigenvalue problem in A third, less frequently used approach is theGram-Schmidt RotationsWe illustrate the process in the case where is a 2 2 matrix. In Gaussian elimination, we compute 1 = where 1is unit lower triangular and is upper triangular.
0:8147 0:0975 0:1576 0:9058 0:2785 0:9706 0:1270 0:5469 0:9572 0:9134 0:9575 0:4854 0:6324 0:9649 0:8003 3 7 7 7 7 5: First, we compute a Givens rotation that, when applied to a 41 and a 51, zeros a 51: 0:8222 0:5692 0:5692 0:8222 T 0:9134 0:6324 = 1:1109 0 : Applying this rotation to rows 4 and 5 yields 2 6 6 6 6 4 1 0 0 0 0 0 1 0 0 0 0 0 1 0 ...
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