v2006.03.09 - Convex Optimization

452 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 (1161)There are η singular values. Rank is equivalent to the number ofnonzero singular values, on the main diagonal of Σ for any flavor SVD,as is well known. Now⎡q T1A = UΣQ T = [ u 1 · · · u ρ ] Σ⎣.⎤∑⎦ = ρ σ i u i qiTqρT i=1(1162)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 )(1163)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 (1160) provides what is called thefull singular value decomposition of A ∈ R m×n [212, 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)(1164)