Lecture_9_1 - Lecture Note 9-1: Nov 7 - Nov 10, 2006 Dr....

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Unformatted text preview: Lecture Note 9-1: Nov 7 - Nov 10, 2006 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong jwong@math.cuhk.edu.hk MAT 2310 Linear Algebra and Its Applications Fall, 2006 Produced by Jeff Chak-Fu WONG 1 A PPLICATIONS OF R EAL V ECTOR S PACES 1. QR-Factorization 2. Least Squares APPLICATIONS OF REAL VECTOR SPACES 2 QR-F ACTORIZATION We discussed the LU-factorization of a matrix and showed how it leads to a very efficient method for solving a linear system. We now discuss another factorization of a matrix A , called the QR-factorization of A . This type of factorization is widely used in computer codes 1. to find the eigenvalues of a matrix (we shall discuss this topic soon), 2. to solve linearly systems, and 3. to find least squares approximations (in this lecture). QR-FACTORIZATION 3 Remark The Gram-Schmidt process with subsequent normalization not only converts an arbitrary basis { u 1 , u 2 , ··· , u n } into an orthonormal basis { w 1 , w 2 , ··· , w n } , but it does it in such a way that for k ≥ 2 the following relationships hold: • { w 1 , w 2 , ··· , w k } is an orthonormal basis for the space spanned by { u 1 , u 2 , ··· , u k } . • w k is orthogonal to { u 1 , u 2 , ··· , u k- 1 } . QR-FACTORIZATION 4 Theorem 0.1 If A is an m × n matrix with linearly independent columns, then A can be factored as A = QR , where Q is an m × n matrix whose columns form an orthonormal basis for the column space of A and R is an n × n nonsingular upper triangular matrix. Proof. Let u 1 , u 2 ,..., u n denote the linearly independent columns of A , which form a basis for the column space of A . By using the Gram-Schmidt process (see Theorem 0.3 in Lecture Note 7-2), we can obtain an orthonormal basis w 1 , w 2 ,..., w n for the column space of A . Recall how this orthonormal basis was obtained. We first constructed an orthogonal basis v 1 , v 2 ,..., v n as follows: v 1 = u 1 and then for i = 2 , 3 ,...,n we have v i = u i- u i · v 1 v 1 · v 1 v 1- u i · v 2 v 2 · v 2 v 2- u i · v i- 1 v i- 1 · v i- 1 v i- 1 . (1) Finally, w i = 1 k v i k v i for i = 1 , 2 , 3 ,...,n . Now each of the vectors u i can be QR-FACTORIZATION 5 written as a linear combination of the w-vectors: u 1 = ( u 1 · w 1 ) w 1 + ( u 1 · w 2 ) w 2 + ··· + ( u 1 · w n ) w n u 2 = ( u 2 · w 1 ) w 1 + ( u 2 · w 2 ) w 2 + ··· + ( u 2 · w n ) w n . . . u n = ( u n · w 1 ) w 1 + ( u n · w 2 ) w 2 + ··· + ( u n · w n ) w n . (2) From Theorem 0.2 (cf. Lecture Note 7-2) we have r ji = u i · w j . Moreover, from Equation (1), we see that u i lies in span { v 1 , v 2 ,..., v i } = span { w 1 , w 2 ,..., w i } . Since w j is orthogonal to span { w 1 , w 2 ,..., w i } for j > i , it is orthogonal to u i ....
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Lecture_9_1 - Lecture Note 9-1: Nov 7 - Nov 10, 2006 Dr....

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