Nonnegative matrices with prescribed extremal singular values

We consider the problem of constructing nonnegative matrices with prescribed extremal singular values. In particular, given 2n−12n−1 real numbers σ1(j) and σj(j),j=1,…,n, we construct an n×nn×n nonnegative bidiagonal matrix BB and an n×nn×n nonnegative semi-bordered diagonal matrix CC, such that σ1(j) and σj(j) are, respectively, the minimal and the maximal singular values of certain submatrices BjBj and CjCj of BB and CC, respectively. By using a singular value perturbation result, we also construct an n×nn×n nonnegative matrix with prescribed singular values σ1≥⋯≥σnσ1≥⋯≥σn.