v2007.09.13 - Convex Optimization

506 APPENDIX A. LINEAR ALGEBRAA.6.2Subcompact SVDSome authors allow only nonzero singular values. In that case the compactdecomposition can be made smaller; it can be redimensioned in terms of rankρ because, for any A∈ R m×nρ = rankA = rank Σ = max {i∈{1... η} | σ i ≠ 0} ≤ η (1351)There are η singular values. For any flavor SVD, rank is equivalent tothe number of nonzero singular values on the main diagonal of Σ.Now⎡q T1A = UΣQ T = [ u 1 · · · u ρ ] Σ⎣.⎤∑⎦ = ρ σ i u i qiTqρT i=1(1352)U ∈ R m×ρ , Σ ∈ R ρ×ρ , Q ∈ R n×ρwhere the main diagonal of diagonal matrix Σ has no 0 entries, andR{u i } = R(A)R{q i } = R(A T )(1353)A.6.3Full SVDAnother common and useful expression of the SVD makes U and Qsquare; making the decomposition larger than compact SVD. Completingthe nullspace bases in U and Q from (1350) provides what is called thefull singular value decomposition of A ∈ R m×n [247, App.A]. Orthonormalmatrices U and Q become orthogonal matrices (B.5):R{u i |σ i ≠0} = R(A)R{u i |σ i =0} = N(A T )R{q i |σ i ≠0} = R(A T )R{q i |σ i =0} = N(A)(1354)