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cs3220p5

# cs3220p5 - A different sort of normal 1 Derive a version of...

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A different sort of normal 1. Derive a version of the normal equations to minimize the M norm Let , then we can define the directional derivative of as where so for t = 0, the original equation becomes 2. Show the 2-norm form equivalence of the M form 3. Given the Cholesky decomposition of M, find the LSR equivalent minimizing the M- norm Which gives us the matlab code LT = chol(M) [Q R] = qr( , 0) x = R \ ( ) SVD Stuff

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1. Show that First, we know that where Since by definition, must be fully orthogonal to , the second term must be 0. Furthermore, we know that , we know that hence , which means that is still orthogonal, hence but earlier we also claimed that , meaning by transitivity of equality. 2. Show that First, because , we know that such that Now let's simplify the LHS (note that )
Finally, let's simplify the RHS so the RHS simplifies down to . Hence 3. Show that First, let's show that . If we recall from lecture, we've shown that is the maximum eigenvalue of . Since is diagonal, its diagonal is by definition the set of eigenvalues. And since

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