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

SVD m x nAny real m x n matrix A can be decomposed uniquely: m x nU is m x n and column orthonormal (U T U=I) n x nD is n x n and diagonal –σ i are called singular values of A –It is assumed that σ 1 ≥ σ 2 ≥ … ≥ σ n ≥ 0 n x nV is n x n and orthonormal (VV T =V T V=I)

SVD (cont’d) m=n,If m=n, then: n x nU is n x n and orthonormal (U T U=UU T =I) n x nD is n x n and diagonal n x nV is n x n and orthonormal (VV T =V T V=I)

SVD (cont’d) The columns of U are eigenvectors of AA T The columns of V are eigenvectors of A T A If λ i is an eigenvalue of A T A (or AA T ), then λ i = σ i 2 for square matrices: A=PΛP -1

SVD - Example D U = (u 1 u 2... u n )V = (v 1 v 2... v n )

SVD – Another Example The eigenvalues of AA T, A T A are: The eigenvectors of AA T, A T A are: λ1λ2λ3λ1λ2λ3

SVD properties A square (n × n) matrix A is singular iff at least one of its singular values σ 1, …, σ n is zero. The rank of matrix A is equal to the number of nonzero singular values σ i

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

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! (U T U=UU T =I or U T =U -1 and V T V=VV T =I or V T =V -1 )

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) ?