cs357-slides-orthog (1)

# cs357-slides-orthog (1) - orthogonalization L Olson...

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orthogonalization L. Olson Department of Computer Science University of Illinois at Urbana-Champaign 1

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objectives Understand the concepts of orthogonal and orthonormal vectors Know the definition of an orthogonal matrix Know how to construct a pair of orthogonal vectors Be able to implement the Gram-Schmidt algorithm Gram-Schmidt orthonormalizes of a set of vectors Understand how to find the QR factorization of a matrix 2
conditioning We have seen A T A used to compute the SVD of a matrix A . We will see A T Ax = A T b used to fit a line to a set of data points. How does the conditioning of A T A compare to that of A ? Theorem cond ( A T A ) = ( cond ( A )) 2 1 A = np.random.rand(10,10) 2 print (np.linalg.cond(A)) 3 print (np.linalg.cond(A.T.dot(A))) 4 50.0972712517 5 2509.73658686 What does the condition number tell us? This is an issue when solving a system of equations... Today we will look at another matrix factorization A = QR An alternative to solving the system A T Ax = A T b 3

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