v2010.10.26 - Convex Optimization

622 APPENDIX A. LINEAR ALGEBRANowq T1A = UΣQ T = [ u 1 · · · u ρ ] Σ⎣.⎡⎤∑⎦ = ρ σ i u i qiTqρT i=1(1570)U ∈ R m×ρ , Σ ∈ R ρ×ρ , Q ∈ R n×ρwhere the main diagonal of diagonal matrix Σ has no 0 entries, andA.6.3Full SVDR{u i } = R(A)R{q i } = R(A T )(1571)Another 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 (1568) provides what is called thefull singular value decomposition of A∈ R m×n [331, 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)For any matrix A having rank ρ (= rank Σ)⎡ ⎤q T1 ∑A = UΣQ T = [ u 1 · · · u m ] Σ⎣. ⎦ = η σ i u i qiTq T n⎡σ 1= [ m×ρ basis R(A) m×m−ρ basis N(A T ) ] σ 2⎢⎣...i=1(1572)⎤⎡ ( n×ρ basis R(A T ) ) ⎤T⎥⎣⎦⎦(n×n−ρ basis N(A)) TU ∈ R m×m , Σ ∈ R m×n , Q ∈ R n×n (1573)where upper limit of summation η is defined in (1564). Matrix Σ is nolonger necessarily square, now padded with respect to (1565) by m−ηzero rows or n−η zero columns; the nonincreasingly ordered (possibly 0)singular values appear along its main diagonal as for compact SVD (1566).