Nonnegativity preservation under singular values perturbation (Q1036322)

Summary: We study how singular values and singular vectors of a matrix \(A\) change, under matrix perturbations of the form \(A+\alpha{\mathbf u}_i{\mathbf v}_i^*\) and \(A+\alpha{\mathbf u}_p{\mathbf v}_{q'}^*\), \(p\neq q\), where \(\alpha\in\mathbb R\), \(A\) is an \(m\times n\) positive matrix with singular values \(\sigma_1\geq\sigma_2\geq\cdots\geq \sigma_r>0\), \(r=\min\{m,n\}\), and \({\mathbf u}_j,{\mathbf v}_k\), \(j=1,\dots,m\); \(k=1,\dots,n\), are the left and right singular vectors, respectively. In particular we give conditions under which this kind of perturbations preserve nonnegativity and certain matrix structures.