Singular Value Decomposition (SVD) Properties

1. Singular Value Decomposition (SVD)(see Appendix A.6, Trucco & Verri) CS485/685 Computer Vision Prof. George Bebis

2. SVD • Any real m x nmatrix A can be decomposed uniquely: • U is m x nand column orthonormal (UTU=I) • D is n x nanddiagonal • σi are called singularvalues of A • It is assumed that σ1 ≥ σ2 ≥ … ≥ σn ≥ 0 • V is n x nand orthonormal (VVT=VTV=I)

3. SVD (cont’d) • If m=n, then: • U is n x nand orthonormal (UTU=UUT=I) • D is n x nanddiagonal • V is n x nand orthonormal (VVT=VTV=I)

4. SVD (cont’d) • The columns of U are eigenvectors of AAT • The columns of V are eigenvectors of ATA • If λi is an eigenvalue of ATA (or AAT), then λi =σi2 for square matrices: A=PΛP-1

5. SVD - Example U = (u1u2 . . . un) V = (v1v2 . . . vn) D

6. λ1 λ2 λ3 SVD – Another Example The eigenvalues of AAT, ATA are: The eigenvectors of AAT, ATA are:

7. SVD properties • A square (n × n) matrix A is singular iff at least one of its singular values σ1, …, σn is zero. • The rank of matrix A is equal to the number of nonzerosingular values σi

8. Matrix “condition” • SVD gives a way of determining how singular A is. • The conditionof Ameasures the degree of singularity of A: cond(A)= • (ratio of largest singular value to its smallest singular value) • Matrices with large condition number are called • ill conditioned.

9. Computing A-1 using SVD • If A is a n x n nonsingular matrix, then its inverse can be computed as follows: easy to compute! (UTU=UUT=I or UT=U-1 and VTV=VVT=I or VT=V-1)

10. Computing A-1 using SVD (cont’d) • If A is singular (or ill-conditioned), we can use SVD to approximate its inverse as follows: ? where (t is a small threshold)