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week5review

# week5review - 3-1 Review Session 5 Review Session 5 Recap...

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3 - 1 Review Session 5 Scott Hsieh, Stanford 2011.05.03.01 Review Session 5 Recap of eigenvalues and eigenvectors The singular value decomposition Practice problems

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3 - 2 Review Session 5 Scott Hsieh, Stanford 2011.05.03.01 Eigenvalues and eigenvectors The fundamental equation for eigenvalues is Av = λv Writing this a series of equations, we get AV = V Λ If V is full rank, then we can invert it to get A = V Λ V - 1 Here, V transforms from eigenvector coordinates to standard R n coordinates. (Do you see why?) V - 1 does the inverse. Moral: in the eigenvector coordinates, matrix operations are easy when the eigenvalue decomposition works.
Review Session 5 Scott Hsieh, Stanford 2011.05.03.01 The Singular Value Decomposition The eigenvalue decomposition gave us A = V Λ V - 1 The singular value decomposition instead gives us A = U Σ V T Here, U and V are orthogonal, and Σ is diagonal. Thus, there is a geometric interpretation, that a unit ball maps to an ellipsoid. The ellipsoid can be interpreted in the context of a controls or estimation problem. Every matrix can be decomposed using the SVD.

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week5review - 3-1 Review Session 5 Review Session 5 Recap...

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