In this note, we establish some connection between the nonnegative inverseeigenvalue problem and that of doubly stochastic one. More precisely, we provethat if $(r; {\lambda}_2, ..., {\lambda}_n)$ is the spectrum of an $n\times n$nonnegative matrix A with Perron eigenvalue r, then there exists a least realnumber $k_A\geq -r$ such that $(r+\epsilon; {\lambda}_2, ..., {\lambda}_n)$ isthe spectrum of an $n\times n$ nonnegative generalized doubly stochastic matrixfor all $\epsilon\geq k_A.$ As a consequence, any solutions for the nonnegativeinverse eigenvalue problem will yield solutions to the doubly stochasticinverse eigenvalue problem. In addition, we give a new sufficient condition fora stochastic matrix A to be cospectral to a doubly stochastic matrix B and inthis case B is shown to be the unique closest doubly stochastic matrix to Awith respect to the Frobenius norm. Some related results are also discussed.
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