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 = 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. The Singular Value DecompositionVT
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 DecompositionVT =
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 DecompositionA 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