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Unformatted text preview: Lecture 20 SVD and Its Applications ShangHua Teng Every symmetric matrix A can be written as Spectral Theorem and Spectral Decomposition [ ] T n n n T T n T n n x x x x x x x x A λ λ λ λ + + = = 1 1 1 1 1 1 where x 1 … x n are the n orthonormal eigenvectors of A, they are the principal axis of A. Singular Value Decomposition • Any m by n matrix A may be factored such that A = U Σ V T • U : m by m , orthogonal, columns • V : n by n , orthogonal, columns 2200 Σ : m by n , diagonal, r singular values The Singular Value Decomposition r = the rank of A = number of linearly independent columns/rows A U V T m x n m x m m x n n x n · · = Σ SVD Properties • U , V give us orthonormal bases for the subspaces of A : – 1st r columns of U : Column space of A – Last m  r columns of U : Left nullspace of A – 1st r columns of V : Row space of A – 1st n r columns of V : Nullspace of A • IMPLICATION: Rank( A ) = r The Singular Value Decomposition · · A U V T = Σ A U V T m x n m x r r x r r x n = Σ m x n m x m m x n n x n Singular Value Decomposition...
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This note was uploaded on 03/08/2010 for the course CS 232 taught by Professor Bera during the Spring '09 term at BU.
 Spring '09
 BERA
 Algorithms

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