On the doubly stochastic realization of spectra

Abstract

An n-list $$\lambda = \left( r; \lambda _2, \ldots , \lambda _n\right)$$ of complex numbers with $$r>\max _{2\le j\le n}|\lambda _j|,$$ is said to be realizable if $$\lambda$$ is the spectrum of $$n\times n$$ nonnegative matrix A and in this case A is said to be a nonnegative realization of $$\lambda .$$ If, in addition, each row and column sum of A is equal to r, then $$\lambda$$ is said to be doubly stochastically realizable and in such case A is said to be a doubly stochastic realization for $$\lambda .$$ In 1997, Guo proved that if $$\left( \lambda _2,\ldots , \lambda _n\right)$$ is any list of complex numbers which is closed under complex conjugation then there exists a least real number $$\lambda _0$$ with $$\max _{2\le j\le n}|\lambda _j|\le \lambda _0\le 2n\max _{2\le j\le n}|\lambda _j|$$ such that the list of complex numbers $$\left( \lambda _1,\lambda _2,\ldots ,\lambda _n\right)$$ is realizable if and only if $$\lambda _1\ge \lambda _0.$$ Many researchers deal with sharpening this upper bound and others are concerned with finding the exact value of $$\lambda _0$$ for particular classes of matrices (Andrade et al. in Linear Algebra Appl 556:301–322, 2018; Andrade et al. in Linear Algebra Appl 551:36–56, 2018; Julio and Soto in Electron J Linear Algebra 36:484–502, 2020; Robbiano in Linear Algebra Appl 564:15–27, 2019). In this paper, we first describe a method for passing from a nonnegative realization to a doubly stochastic realization. As applications, we give a new sufficient condition for a stochastic matrix A to be cospectral to a doubly stochastic matrix B and in this case B is shown to be the unique closest doubly stochastic matrix to A with respect to the Frobenius norm. Then, our next result gives an improvement of Guo’s bound which also sharpens the existing known bound for the case when one of at least one of $$\left\{ \lambda _2, \ldots , \lambda _n\right\}$$ is real. Furthermore, we investigate the case when $$\left\{ \lambda _2, \ldots , \lambda _n\right\}$$ are all non-real which has not been dealt before. Our main results here also sharpen Guo’s bound. Next, for doubly stochastic realizations, we obtain an upper bound that improves Guo’s bound as well. Finally, for certain particular cases, we give a further improvement of our last bound for doubly stochastic realization.