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Unformatted text preview: IE 495 Lecture 22 November 16, 2000 Reading for This Lecture Primary ¡ Miller and Boxer, Pages 128134 ¡ Forsythe and Mohler, Sections 9 and 10 Solving Systems of Equations Problem : Given a matrix A ∈ R n × n and a vector b ∈ R n , we wish to find x ∈ R n such that Ax = b . Diagonal form of a matrix ¡ An othogonal matrix U has the property the U T U = UU T = I . ¡ Given A ∈ R n × n , there exist orthogonal matrices U, V such that ¢ U T AV = D where D is a diagonal matrix where ¢ diagonal elements of D are μ 1 ≥ μ 2 ≥ £ ≥ μ r μ r+1 = ¤ £ μ n = 0 , and ¢ r is the rank of A . ¢ μ i is the nonnegative square root of the i th eigenvalue . ¡ This is called the singular value decomposition . Importance of the SVD Effect of multiplying by a matrix Implications Multiplying by A represents a rotation and a scaling of axes to get from one space to the other. μ i is the nonnegative square root of the i th eigenvalue. Notice that ¡ A ¡ = ¡ D ¡ = μ 1 ....
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This note was uploaded on 08/06/2008 for the course IE 495 taught by Professor Linderoth during the Fall '08 term at Lehigh University .
 Fall '08
 Linderoth
 Operations Research

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