MAT211_Lecture15[1]

MAT211_Lecture15[1] - properties Moreover r 11 = || v 1| ||...

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MAT211 Lecture 15 Gram-Schmidt Process ! The Gram-Schmidt process ! QR Factorization 1 The Gram-Schmidt Process EXAMPLE Perform the Gram-Schimdt process on the sequence of vectors (1,1,1),(2,0,2),(-1,0,-1) Theorem (QR Factorization) Consider an n x n matrix M with linearly independent columns v 1 , v 2 ,.. v n . Then there exists an n x n matrix Q whose columns u 1 , u 2 ,.. u n are orthonormal and an upper triangular matrix R with positive diagonal entries such that M = Q R. The matrices Q and R are unique with the above
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Unformatted text preview: properties. Moreover, r 11 = || v 1| || , r jj = || v j ! ||for j=2. .n, and r ij = u i . v j for i< j . Theorem (QR Factorization Algorithm) Consider an n x n matrix M with linearly independent columns v 1 , v 2 ,.. v n . Then the columns q 1 , q 2 ,.. q n of Q and the columns of R can be computed in the following order First col of R, ±rst column of U Second col of R, second col of U and so on EXAMPLE: Find the QR factorization of the matrix 1 2 1 1 1 1 1 2 1 1 1...
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