the complexity of divisibility we address two sets of long standing open questions in linear algebra and probability theory from a computational complexity perspective stochastic matrix divisibility and divisibility and decomposability of probability distributions we prove that finite divisibility of stochastic matrices is an np complete problem and extend this result to nonnegative matrices and completely positive trace preserving maps i e the quantum analogue of stochastic matrices we further prove a complexity hierarchy for the divisibility and decomposability of probability distributions showing that finite distribution divisibility is in p but decomposability is np hard for the former we give an explicit polynomial time algorithm all results on distributions extend to weak membership formulations proving that the complexity of these problems is robust to perturbations

exit problems associated with affine reflection groups we obtain a formula for the distribution of the first exit time of brownian motion from the alcove of an affine weyl group in most cases the formula is expressed compactly in terms of pfaffians expected exit times are derived in the type widetilde a case the results extend to other markov processes we also give formulas for the real eigenfunctions of the dirichlet and neumann laplacians on alcoves observing that the hot spots conjecture of j rauch is true for alcoves

on the symmetric doubly stochastic inverse eigenvalue problem abstract let 1 2 n be a list of real numbers the symmetric doubly stochastic inverse eigenvalue problem hereafter sdiep is to determine the list which can occur as the spectrum of an n n symmetric doubly stochastic matrix a if the matrix a is positive we can necessarily obtain a subproblem symmetric positive doubly stochastic inverse eigenvalue problem hereafter spdiep of the sdiep in this paper we give some sufficient conditions for the sdiep and spdiep and prove that the set formed by the spectra of all n n symmetric positive doubly stochastic matrices is non convex for n 4

a necessary condition for the spectrum of nonnegative symmetric 5 5 matrices abstract let a be a nonnegative symmetric 5 5 matrix with eigenvalues 1 2 3 4 5 we show that if i 1 5 i 1 2 1 then 3 i 1 5 i mcdonald and neumann showed that 1 3 4 0 let 1 2 3 4 5 be a list of decreasing real numbers satisfying 1 i 1 5 i 1 2 1 2 3 i 1 5 i 3 1 3 4 0 4 the perron property that is 1 max we show that is the spectrum of a nonnegative symmetric 5 5 matrix thus we solve the symmetric nonnegative inverse eigenvalue problem for n 5 in a region for which a solution has not been known before

realizing suleĭmanova spectra via permutative matrices ii abstract in this work the real nonnegative inverse eigenvalue problem is solved for a particular class of permutative matrix the necessary and sufficient condition there is also shown to be sufficient for the symmetric nonnegative inverse eigenvalue problem a result due to johnson and paparella 2016 8 is extended to include normalized lists that satisfy the new sufficient condition

algebraic shift equivalence and primitive matrices motivated by symbolic dynamics we study the problem given a unital subring s of the reals when is a matrix a algebraically shift equivalent over s to a primitive matrix we conjecture that simple necessary conditions on the nonzero spectrum of a are sufficient and establish the conjecture in many cases if s is the integers we give some lower bounds on sizes of realizing primitive matrices for dedekind domains we prove that algebraic shift equivalence implies algebraic strong shift equivalence

a map of sufficient conditions for the real nonnegative inverse eigenvalue problem abstract the real nonnegative inverse eigenvalue problem rniep is the problem of determining necessary and sufficient conditions for a list of real numbers to be the spectrum of an entrywise nonnegative matrix a number of sufficient conditions for the existence of such a matrix are known in this paper in order to construct a map of sufficient conditions we compare these conditions and establish inclusion relations or independency relations between them

a unified view on compensation criteria in the real nonnegative inverse eigenvalue problem a connection is established between the problem of characterizing all possible real spectra of entrywise nonnegative matrices the so called real nonnegative inverse eigenvalue problem and a combinatorial process consisting in repeated application of three basic manipulations on sets of real numbers given realizable sets i e sets which are spectra of some nonnegative matrix each of these three elementary transformations constructs a new realizable set this defines a special kind of realizability called c realizability and this is closely related to the idea of compensation after observing that the set of all c realizable sets is a strict subset of the set of realizable ones we show that it strictly includes in particular all sets satisfying several previously known sufficient realizability conditions in the literature furthermore the proofs of these conditions become much simpler when approached from this new point of view

semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata in this paper we study the semidefinite inverse eigenvalue problem of reconstructing a real n by n matrix c such that it is nearest to the original pre estimated real n by n matrix c o in the frobenius norm and satisfies the measured partial eigendata where the required matrix c should preserve the symmetry positive semidefiniteness and the prescribed entries of the pre estimated matrix c o we propose the alternating direction method of multipliers for solving the semidefinite inverse eigenvalue problem where three related iterative algorithms are presented we also extend our method to the case of lower bounds numerical experiments are reported to illustrate the efficiency of the proposed method for solving semidefinite inverse eigenvalue problems

on the terwilliger algebra of distance biregular graphs abstract let denote a distance biregular graph with vertex set x fix x x and let t t x denote the terwilliger algebra of with respect to x in this paper we consider irreducible t modules with endpoint 1 we show that there are no such modules if and only if is the complete bipartite graph k 1 n n 1 and x is a vertex of with valency 1 if the valency of x is at least 2 then we show that up to isomorphism there is a unique irreducible t module of endpoint 1 and this module is thin

elgenvalues of nonnegative matrices abstract in a partial continuation of work by fiedler some spectral properties of symmetric nonnegative matrices are extended to general nonnegative matrices

nonnegative realization of spectra having negative real parts abstract the nonnegative inverse eigenvalue problem niep is the problem of determining necessary and sufficient conditions for a list of complex numbers to be the spectrum of a nonnegative matrix in this paper the problem is completely solved in the case when all numbers in the given list except for one the perron eigenvalue have real parts smaller than or equal to zero

realizability criterion for the symmetric nonnegative inverse eigenvalue problem abstract let 1 2 n a set of real numbers the real nonnegative inverse eigenvalue problem rniep is the problem of determining necessary and sufficient conditions in order that be the spectrum of an entrywise nonnegative n n matrix if there exists a nonnegative matrix a with spectrum we say that is realized by a many realizability criteria for the existence of such a matrix a are known this paper shows that a realizability criterion given by the author which contains both kellogg s realizability criterion and borobia s realizability criterion is sufficient for the existence of an n n symmetric nonnegative matrix with prescribed spectrum

on the algebraic theory of pseudo distance regularity around a set abstract let be a connected graph with vertex set v adjacency matrix a positive eigenvector and corresponding eigenvalue 0 a natural generalization of distance regularity around a vertex subset c v which makes sense even with non regular graphs is studied this new concept is called pseudo distance regularity and its definition is based on giving to each vertex u v a weight which equals the corresponding entry u of and regularizes the graph this approach reveals a kind of central symmetry which in fact is an inherent property of all kinds of distance regularity because of the distance partition of v they come from we come across such a concept via an orthogonal sequence of polynomials constructed from the local spectrum of c called the adjacency polynomials because their definition strongly relies on the adjacency matrix a in particular it is shown that c is tight that is the corresponding adjacency polynomials attain their maxima at 0 if and only if is pseudo distance regular around c as an application some new spectral characterizations of distance regularity around a set and completely regular codes are given

a characterization of trace zero nonnegative 5 5 matrices let 1 2 3 4 5 be a list of complex numbers with 1 2 3 4 5 0 necessary and sufficient conditions for the existence of an entry wise nonnegative 5 5 matrix a with spectrum are presented

pseudo distance regularized graphs are distance regular or distance biregular abstract the concept of pseudo distance regularity around a vertex of a graph is a natural generalization for non regular graphs of the standard distance regularity around a vertex in this note we prove that a pseudo distance regular graph around each of its vertices is either distance regular or distance biregular by using a combinatorial approach the same conclusion was reached by godsil and shawe taylor for a distance regular graph around each of its vertices thus our proof which is of an algebraic nature can also be seen as an alternative demonstration of godsil and shawe taylor s theorem

on pseudo distance regularity abstract the concept of local pseudo distance regularity recently introduced is a natural generalization of distance regularity intended for not necessarily regular graphs we study here some properties of locally pseudo distance regular graphs and give some new characterization of such structures as a consequence some new characterizations of distance regular graphs are also derived

on the alternating direction method of multipliers for nonnegative inverse eigenvalue problems with partial eigendata we consider the nonnegative inverse eigenvalue problem with partial eigendata which aims to find a nonnegative matrix such that it is nearest to a pre estimated nonnegative matrix and satisfies the prescribed eigendata in this paper we propose several iterative schemes based on the alternating direction method of multipliers for solving the nonnegative inverse problem we also extend our schemes to the symmetric case and the cases of prescribed lower bounds and of prescribed entries numerical tests including a practical engineering application in vibrations show the efficiency of the proposed iterative schemes

perron spectratopes and the real nonnegative inverse eigenvalue problem abstract call an n by n invertible matrix s a perron similarity if there is a real non scalar diagonal matrix d such that s d s 1 is entrywise nonnegative we give two characterizations of perron similarities and study the polyhedra c s x r n s d x s 1 0 d x diag x and p s x c s x 1 1 which we call the perron spectracone and perron spectratope respectively the set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by perron spectratopes so that enumerating them is of interest the perron spectracone and spectratope of hadamard matrices are of particular interest and tend to have large volume for the canonical hadamard matrix as well as other matrices the perron spectratope coincides with the convex hull of its rows in addition we provide a constructive version of a result due to fiedler 9 theorem 2 4 for hadamard orders and a constructive version of the boyle handelman theorem 2 theorem 5 1 for sule manova spectra

realizable lists on a class of nonnegative matrices abstract a square matrix of order n with n 2 is called permutative matrix when all its rows are permutations of the first row in this paper recalling spectral results for partitioned into 2 by 2 symmetric blocks matrices sufficient conditions on a given complex list to be the list of the eigenvalues of a nonnegative permutative matrix are given in particular we study niep and pniep when some complex elements in the lists under consideration have non zero imaginary part realizability regions for nonnegative permutative matrices are obtained a guo s realizability preserving perturbations result is obtained

realizable lists via the spectra of block matrices abstract in this paper we present spectral results for matrices partitioned into higher order blocks where not all blocks are necessarily square using these results sufficient conditions on a given list to be the list of eigenvalues of a nonnegative matrix are obtained and the corresponding matrix is constructed in particular spectral results for symmetric matrices are derived a result on guo perturbations for permutative matrices on given lists is generalized

an algorithm for constructing nonnegative matrices with prescribed real eigenvalues provided with the real spectrum this paper presents a numerical procedure based on the induction principle to solve two types of inverse eigenvalue problems one for nonnegative matrices and another for symmetric nonnegative matrices as an immediate application our approach can offer not only the sufficient condition to solve inverse eigenvalue problems for nonnegative or symmetric nonnegative matrices but also a numerical way to solve inverse eigenvalue problems for stochastic matrices numerical examples are presented to show the capacity of our method

the nonnegative inverse eigenvalue problem abstract let 1 n be a list of complex numbers the nonnegative inverse eigenvalue problem niep is the problem of determining necessary and sufficient conditions in order that be the spectrum of an entrywise nonnegative n n matrix our purpose is to give an overview of the niep its history modern results and subproblems with particular emphasis on symmetric realizing matrices a substantial bibliography is included

on the nonnegative eigenvalue problem abstract 1 1 n is the spectrum of a stochastic matrix of order n 1 if and only if there exists some real matrix b of order n with spectrum 1 n and some n simplex s r n containing the origin such that bs s this result appears implicitly in ciarlet s work and it provides us with a geometrical tool to obtain sufficient conditions for the nonnegative eigenvalue problem we employ it here to generalize the sufficient conditions given by kellogg

a geometrical representation of the spectra of four dimensional nonnegative matrices abstract a geometrical representation of the set of four complex numbers which are the spectrum of 4 dimensional entrywise nonnegative real matrices is provided the characterization is based on the result for the nonnegative inverse eigenvalue problem niep from the coefficients of the characteristic polynomial given in 16

nonnegative inverse eigenvalue problems with partial eigendata in this paper we consider the inverse problem of constructing an n n real nonnegative matrix a from the prescribed partial eigendata we first give the solvability conditions for the inverse problem without the nonnegative constraint and then discuss the associated best approximation problem to find a nonnegative solution we reformulate the inverse problem as a monotone complementarity problem and propose a nonsmooth newton type method for solving its equivalent nonsmooth equation under some mild assumptions the global and quadratic convergence of our method is established we also apply our method to the symmetric nonnegative inverse problem and to the cases of prescribed lower bounds and of prescribed entries numerical tests demonstrate the efficiency of the proposed method and support our theoretical findings

isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure the nonnegative inverse eigenvalue problem is that given a family of complex numbers l l 1 l n find a nonnegative matrix of order n with spectrum l this problem is difficult and remains unsolved partially in this paper we focus on its generalization that the reconstructed nonnegative matrices should have some prescribed entries it is easy to see that this new problem will come back to the common nonnegative inverse eigenvalue problem if there is no constraint of the locations of entries a numerical isospectral flow method which is developed by hybridizing the optimization theory and steepest descent method is used to study the reconstruction moreover an error estimate of the numerical iteration for ordinary differential equations on the matrix manifold is presented after that a numerical method for the nonnegative symmetric inverse eigenvalue problem with prescribed entries and its error estimate are considered finally the approaches are verified by the numerical test results

an algorithm for constructing doubly stochastic matrices for the inverse eigenvalue problem abstract in this note we present an algorithm that yields a new method for constructing doubly stochastic and symmetric doubly stochastic matrices for the inverse eigenvalue problem in addition we introduce new open problems in this area that lay the ground for future work

on the comparison of some realizability criteria for the real nonnegative inverse eigenvalue problem abstract a result by brauer shows how to modify one single eigenvalue of a matrix via a rank one perturbation without changing any of the remaining eigenvalues this together with the properties of real matrices with constant row sums was exploited by the authors in a previous work in connection with the nonnegative inverse eigenvalue problem obtaining conditions which are sufficient for the existence of an entrywise nonnegative matrix with prescribed spectrum in this work we make use of brauer s theorem again to show that most of the previous results giving sufficient conditions for the real nonnegative inverse eigenvalue problem can be derived by using brauer s theorem moreover the technique is constructive and there is an algorithmic procedure to construct a matrix realizing the spectrum in particular we show that if either kellogg s realizability criterion or borobia s realizability criterion is satisfied then soto s realizability criterion is also satisfied none of the converses are true thus the condition given by soto appears to be the most general sufficient condition so far for the real nonnegative inverse eigenvalue problem

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

the inverse eigenvalue problem for entanglement witnesses abstract we consider the inverse eigenvalue problem for entanglement witnesses which asks for a characterization of their possible spectra or equivalently of the possible spectra resulting from positive linear maps of matrices we completely solve this problem in the two qubit case and we derive a large family of new necessary conditions on the spectra in arbitrary dimensions we also establish a natural duality relationship with the set of absolutely separable states and we completely characterize witnesses i e separating hyperplanes of that set when one of the local dimensions is 2

on the realizability of the critical points of a realizable list abstract the nonnegative inverse eigenvalue problem niep is to characterize the spectra of entrywise nonnegative matrices a finite multiset of complex numbers is called realizable if it is the spectrum of an entrywise nonnegative matrix monov conjectured that the kth moments of the list of critical points of a realizable list are nonnegative johnson further conjectured that the list of critical points must be realizable in this work johnson s conjecture and consequently monov s conjecture is established for a variety of important cases including ciarlet spectra sule manova spectra spectra realizable via companion matrices and spectra realizable via similarity by a complex hadamard matrix additionally we prove a result on differentiators and trace vectors and use it to provide an alternative proof of a result due to malamud and a generalization of a result due to kushel and tyaglov on circulant matrices implications for further research are discussed

ruling out certain 5 spectra for the symmetric nonnegative inverse eigenvalue problem abstract a method is developed to show that certain spectra cannot be realized for the s niep it is applied in the 5 by 5 case to rule out many spectra that were previously unresolved these are all in the case of 3 positive and 2 negative eigenvalues as all other cases are now resolved for spectra of the sort we discuss a diagram is given of the spectra that are excluded here as well as those trivially realizable those realizable because of the trace 0 case and those that may also be excluded because of the j l l conditions a small region remains unresolved it is a very small fraction of the area of those spectra we consider

construction of nonnegative symmetric matrices with given spectrum abstract let 1 n be the spectrum of a nonnegative symmetric matrix a with the perron eigenvalue 1 a diagonal entry c and let 1 m be the spectrum of a nonnegative symmetric matrix b with the perron eigenvalue 1 we show how to construct a nonnegative symmetric matrix c with the spectrum 1 max 0 1 c 2 n 2 m

applications of a brauer theorem in the nonnegative inverse eigenvalue problem abstract a result by brauer which shows how to modify one single eigenvalue of a matrix without changing any of the remaining eigenvalues plays a relevant role in the study of the nonnegative inverse eigenvalue problem niep perfect in a long time ignored paper 1955 presents an extension of this result which shows how to modify r eigenvalues of a matrix of order n r n via a rank r perturbation without changing any of the n r remaining eigenvalues by using this extension perfect gives a realizability criterion for the real niep which is not contained in soto s realizability criterion in this work by extending perfect s result we give a new realizability criterion for the real niep which contains soto s criterion thus this new realizability criterion appears to be the most general sufficient condition for the real niep so far we also contribute to the solution of the symmetric niep for n 5

perturbing eigenvalues of nonnegative matrices abstract let a be an irreducible entrywise nonnegative n n matrix with eigenvalues 2 b i c 3 b i c 4 n where is the perron eigenvalue it is shown that for any t 0 there is a nonnegative matrix with eigenvalues t 2 t 3 t 4 n whenever t n t with 3 1 4 2 5 5 and n 2 25 for n 6 the result improves that of guo et al our proof depends on an auxiliary result in geometry asserting that the area of an n sided convex polygon is bounded by n times the maximum area of a triangle lying inside the polygon

guo perturbation for symmetric nonnegative circulant matrices abstract guo w guo eigenvalues of nonnegative matrices linear algebra appl 266 1997 261 270 sets the question if the list 1 2 n is symmetrically realizable that is is the spectrum of a symmetric nonnegative matrix and t 0 whether or not the list t 1 t 2 t 3 n is also symmetrically realizable in this paper we give an affirmative answer to this question in the case that the realizing matrix is circulant or left circulant we also give a necessary and sufficient condition for to be the spectrum of a nonnegative left circulant matrix

an inverse eigenvalue problem for symmetric and normal matrices abstract some new sufficient conditions are found for n real numbers to be the spectrum of some n n symmetric nonnegative matrix and for n complex numbers to be the spectrum of some n n normal nonnegative matrix