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# lecture19 - Lecture 19 Singular Value Decomposition...

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Lecture 19 Singular Value Decomposition Shang-Hua Teng

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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 + + = = 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.
Matrix Decomposition Does every matrix, not necessarily square matrix, have a similar decomposition? How can we use such a decomposition?

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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
Singular Value Decomposition [ ] T T T T T T T T v u v u v u v v v v v u u u A 3 3 3 2 2 2 1 1 1 5 4 3 2 1 3 2 1 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 σ + + = =

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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.

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lecture19 - Lecture 19 Singular Value Decomposition...

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