Download presentation

Presentation is loading. Please wait.

1
**Lecture 19 Singular 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 = UVT U: m by m, orthogonal, columns V: n by n, orthogonal, columns : m by n, diagonal, r singular values

5
**Singular Value Decomposition**

6
**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

7
**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

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**

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

10
**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

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 (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

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

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google