v2006.03.09 - Convex Optimization

A.6. SINGULAR VALUE DECOMPOSITION, SVD 455Expressions for U , Q , and Σ follow readily from (1165),AA T U = UΣΣ T and A T AQ = QΣ T Σ (1167)demonstrating that the columns of U are the eigenvectors of AA T and thecolumns of Q are the eigenvectors of A T A. −Neil Muller et alii [170]A.6.4SVD of symmetric matricesA.6.4.0.1 Definition. Step function. (confer6.3.2.0.1)Define the signum-like entrywise vector-valued function ψ : R n → R n thattakes the value 1 corresponding to an entry-value 0 in the argument:[ψ(a) =∆limx i →a ix i|x i | = { 1, ai ≥ 0−1, a i < 0 , i=1... n ]∈ R n (1168)Eigenvalue signs of a symmetric matrix having diagonalizationA = SΛS T (1153) can be absorbed either into real U or real Q from the fullSVD; [226, p.34] (conferC.5.2.1)orA = SΛS T = Sδ(ψ(δ(Λ))) |Λ|S T ∆ = U ΣQ T ∈ S n (1169)A = SΛS T = S|Λ|δ(ψ(δ(Λ)))S T ∆ = UΣ Q T ∈ S n (1170)where |Λ| denotes entrywise absolute value of diagonal matrix Λ .A.6.5Pseudoinverse by SVDMatrix pseudoinverse (E) is nearly synonymous with singular valuedecomposition because of the elegant expression, given A = UΣQ TA † = QΣ †T U T ∈ R n×m (1171)that applies to all three flavors of SVD, where Σ † simply inverts nonzeroentries of matrix Σ .Given symmetric matrix A∈ S n and its diagonalization A = SΛS T(A.5.2), its pseudoinverse simply inverts all nonzero eigenvalues;A † = SΛ † S T (1172)△