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Lecture 19 Singular Value Decomposition

Lecture 19 Singular Value Decomposition. Shang-Hua Teng. Spectral Theorem and Spectral Decomposition. Every symmetric matrix A can be written as. where x 1 … x n are the n orthonormal eigenvectors of A, they are the principal axis of A.

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Lecture 19 Singular Value Decomposition

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  1. Lecture 19Singular Value Decomposition Shang-Hua Teng

  2. Spectral Theorem and Spectral Decomposition Every symmetric matrix A can be written as where x1 …xn are the n orthonormal eigenvectors of A, they are the principal axis of A. xi xiT is the projection matrix on to xi !!!!!

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

  4. Singular Value Decomposition • Any m by n matrix A may be factored such that A = UVT • U: m by m, orthogonal, columns • V: n by n, orthogonal, columns • : m by n, diagonal, r singular values

  5. Singular Value Decomposition

  6. · · 0 A U S VT = 0 m x n m x m m x n n x n The Singular Value Decomposition r = the rank of A = number of linearly independent columns/rows

  7. The Singular Value Decomposition A U S VT 0 = 0 m x n m x m m x n n x n r = the rank of A = number of linearly independent columns/rows

  8. 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

  9. · · The Singular Value Decomposition 0 A U S VT = 0 m x n m x m m x n n x n 0 A U S VT 0 = m x n m x r r x r r x n

  10. A U S VT 0 = 0 m x n m x m m x n n x n A U S VT 0 = 0 m x n m x r r x r r x n The Singular Value Decomposition

  11. Singular Value Decomposition • where • u1 …ur are the r orthonormal vectors that are basis of C(A) and • v1 …vr are the r orthonormal vectors that are basis of C(AT )

  12. Matlab Example >> A = rand(3,5)

  13. Matlab Example >> [U,S,V] = svd(A)

  14. SVD Proof • Any m x n matrix A has two symmetric covariant matrices (m x m) AAT (n x n) ATA

  15. Spectral Decomposition of Covariant Matrices • (m x m) AAT =U L1 UT • U is call the left singular vectors of A • (n x n)ATA = V L2 VT • V is call the right singular vectors of A • Claim: are the same

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