Non-negative Matrix Inverse Eigenvalue Problem

Let σ={λ1,λ2,...,λn} be a set of n-tuples with (?)={λ2,...,λn} being closed un-der complex conjugation. The nonnegative inverse eigenvalue problem (hereafter NIEP) is to determine the necessary and sufficient conditions in order that a be the spectrum of an entrywise nonnegative n x n matrix. Examples of widespread applications include the Leontief input-output analysis, finite Markov chains, linear complementary problem-s, and so on. Especially, the emergence of the Perron-Frobenius theorem has made great progress for the NIEP. At present, the problem has been extended to the following problems: NIEP. real nonnegative inverse eigenvalue problem (hereafter RNIEP). symmetric nonneg-ative inverse eigenvalue problem (hereafter SNIEP), stochastic inverse eigenvalue problem (hereafter StIEP), doubly stochastic inverse eigenvalue problem (hereafter DIEP), symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP). My thesis mainly study the solvability of some subproblems of the NIEP. It is organized as the following four chapter-s. Chapter1mainly introduces the research background and significance of the NIEP and presents its current research contents and research progress.In Chapter2, we discuss the solvabilities of the generalized row stochastic matrix. Furthermore, some sufficient conditions, such that a set of n-tuples can be realized by some nonnegative matrix, are also given. Finally, the feasibility of these conditions are verified.Then we study the nonnegative interlacing inverse eigenvalue problem in Chapter3. Two kinds of special Hermitian matrices (tridiagonal plus paw form matrices) are presented. We analyze the properties of the minimal and the maximal eigenvalues of all the principal submatriccs. Afterwards, according to these properties we derive the necessary and sufficient conditions for the interlacing inverse eigenvalue problems of the two kinds of matrices.In Chapter4, we analyze the solvability of the SDIEP. On the one hand, we present the sufficient conditions for the n×n symmetric doubly stochastic matrices for n even. On the other hand, we compare all the known sufficient conditions for the SDIEP and obtain the inclusion relations and intersection relations between them. Finaly, we come up with the symmetric positive doubly stochastic inverse eigenvalue problem (hereafter SPDIEP) and discuss the convexity of the set formed by the spectra of all n x n symmetric positive doubly stochastic matrices.