Nonnegative matrices with prescribed extremal singular values

Abstract

We consider the problem of constructing nonnegative matrices with prescribed extremal singular values. In particular, given $2n-1$ real numbers ${\sigma }_1^{\left(j\right)}$ and ${\sigma }_j^{\left(j\right)},j=1,\dots ,n$, we construct an $n\times n$ nonnegative bidiagonal matrix $B$ and an $n\times n$ nonnegative semi-bordered diagonal matrix $C$, such that ${\sigma }_1^{\left(j\right)}$ and ${\sigma }_j^{\left(j\right)}$ are, respectively, the minimal and the maximal singular values of certain submatrices $B_j$ and $C_j$ of $B$ and $C$, respectively. By using a singular value perturbation result, we also construct an $n\times n$ nonnegative matrix with prescribed singular values ${\sigma }_1\ge \cdots \ge {\sigma }_n$.