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**Lecture 19 Singular Value Decomposition**

Shang-Hua Teng

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**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 !!!!!

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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 = UVT U: m by m, orthogonal, columns V: n by n, orthogonal, columns : m by n, diagonal, r singular values

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**Singular Value Decomposition**

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**The Singular Value Decomposition**

VT m x n m x m m x n n x n = S r = the rank of A = number of linearly independent columns/rows

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**The Singular Value Decomposition**

VT = m x n m x m m x n n x n r = the rank of A = number of linearly independent columns/rows

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**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

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**The Singular Value Decomposition**

A U S VT = m x n m x m m x n n x n A U S VT = m x n m x r r x r r x n

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**The Singular Value Decomposition**

= S VT m x n m x m m x n n x n A U = S VT m x n m x r r x r r x n

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**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 )

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

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

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**SVD Proof (m x m) AAT (n x n) ATA**

Any m x n matrix A has two symmetric covariant matrices (m x m) AAT (n x n) ATA

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**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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