A note on eigenvalues location for trace zero doubly stochastic matrices

Some results on the location of the eigenvalues of trace zero doubly stochastic matrices are provided. A result similar to that provided in [H. Perfect and L. Mirsky. Spectral properties of doubly–stochastic matrices. Monatshefte f¨ur Mathematik, 69(1):35–57, 1965.] for doubly stochastic matrices is given

The NIEP

The nonnegative inverse eigenvalue problem (NIEP) asks which lists of $n$ complex numbers (counting multiplicity) occur as the eigenvalues of some $n$-by-$n$ entry-wise nonnegative matrix. The NIEP has a long history and is a known hard (perhaps the hardest in matrix analysis?) and sought after problem. Thus, there are many subproblems and relevant results in a variety of directions. We survey most work on the problem and its several variants, with an emphasis on recent results, and include 130 references. The survey is divided into: a) the single eigenvalue problems; b) necessary conditions; c) low dimensional results; d) sufficient conditions; e) appending 0's to achieve realizability; f) the graph NIEP's; g) Perron similarities; and h) the relevance of Jordan structure

The Doubly Stochastic Single Eigenvalue Problem: An Empirical Approach

The doubly stochastic single eigenvalue problem asks what is the set DSn of all complex numbers that occur as an eigenvalue of an n-by-n doubly stochastic matrix. For n \u3c 5, this set is known and for the analogous set for (singly) stochastic matrices, the set is known for all n. For Pk , the polygon formed by the k-th roots of unity, Unk=1 Pk ⊆ DSn, as is easily shown. For n \u3c 5, this containment is an equality, but for n = 5, the containment is strict (though it is close). Presented here is substantial, computational evidence that the containment is an equality for 6 ≤ n ≤ 10 and for what DS5 actually is

Inverse problems for symmetric doubly stochastic matrices whose Suleĭmanova spectra are bounded below by 1/2

A new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix is presented; this is a contribution to the classical spectral inverse problem for symmetric doubly stochastic matrices that is still open in its full generality. It is proved that whenever $\lambda_2, \ldots, \lambda_n$ are non-positive real numbers with $1 + \lambda_2 + \ldots + \lambda_n \geqslant 1/2$, then there exists a symmetric, doubly stochastic matrix whose spectrum is precisely $(1, \lambda_2, \ldots, \lambda_n)$. We point out that this criterion is incomparable to the classical sufficient conditions due to Perfect-Mirsky, Soules, and their modern refinements due to Nader et al. We also provide some examples and applications of our results.Comment: Accepted to Linear Algebra and Its Applications, pages 1

Combinatorics and Geometry of Transportation Polytopes: An Update

A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure

A Short Note on Extreme Points of Certain Polytopes

We give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly substochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs