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

on single and double soules matrices abstract until now the concept of a soules basis matrix of sign pattern n consisted of an orthogonal matrix r r n n generated in a certain way from a positive n vector which has the property that for any diagonal matrix diag 1 n with 1 n 0 the symmetric matrix a r r t has nonnegative entries only in the present paper we introduce the notion of a pair of double soules basis matrices of sign pattern n which is a pair of matrices p q each in r n n which are not necessarily orthogonal and which are generated in a certain way from two positive vectors but such that pq t i and such that for any of the aforementioned diagonal matrices the matrix a p q t also has nonnegative entries only we investigate the interesting properties which such matrices a have as a preamble to the above investigation we show that the iterates a k r k r k t generated in the course of the qr algorithm when it is applied to a r r t where r is a soules basis matrix of sign pattern n are again symmetric matrices generated by the soules basis matrices r k of sign pattern n which are themselves modified as the algorithm progresses our work here extends earlier works by soules and elsner et al

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

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

the classification of some generalised bunce deddens algebras abstract we use k theory to prove an isomorphism theorem for a large class of generalised bunce deddens algebras constructed by kribs and solel from a directed graph e and a sequence of positive integers in particular we compute the torsion free component of the k 0 group for a class of generalised bunce deddens algebras to show that supernatural numbers are a complete invariant for this class

legendre spectral methods for the grad div operator this paper describes two legendre spectral methods for the grad div eigenvalue problem in r 2 the first method uses a single grid resulting from the p n circle times p n discretization in primal and dual variational formulations as is well known this method is unstable and exhibits spectral pollution effects increased number of singular eigenvalues and increased multiplicity of some eigenvalues belonging to the regular spectrum our study aims at the understanding of these effects the second spectral method is based on a staggered grid of the p n circle times pn 1 discretization this discretization leads to a stable algorithm free of spurious eigenmodes and with spectral convergence of the regular eigenvalues eigenvectors towards their analytical values in addition divergence free vector fields with sufficient regularity properties are spectrally projected onto the discrete kernel of grad div a clear indication of the robustness of this algorithm c 2007 elsevier b v all rights reserved

orthogonal bases that lead to symmetric nonnegative matrices abstract in a paper dating back to 1983 soules constructs from a positive vector x an orthogonal matrix r which has the property that for any nonnegative diagonal matrix with nonincreasing diagonal entries the matrix r rt has all its entries nonnegative independently fiedler in 1988 showed that any symmetric irreducible nonsingular matrix whose powers are all m matrices and hence an mma matrix in the language of friedland hershkowitz and schneider must have an orthogonal matrix of eigenvectors r which has similar properties to those of r here for a given positive n vector x we investigate the structure of all orthogonal matrices r for which for any nonnegative diagonal matrix as above the matrices r rt are nonnegative up to a permutation of its columns each such r corresponds to a binary tree whose vertices are subsets of the set 1 2 n with the property that each vertex has either no successor or exactly two disjoint successors for such orthogonal matrices r and such nonsingular diagonal matrices we show that the set of matrices of the form r rt and the set of inverse mma matrices i e matrices whose inverses are mma matrices coincide using this result we establish a relation between strictly ultrametric matrices and inverse mma matrices finally we show that the qr factorization of r rt for certain such r s has a special sign pattern

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

on a spectral property of doubly stochastic matrices and its application to their inverse eigenvalue problem abstract in this note we present a useful theorem concerning the spectral properties of doubly stochastic matrices as applications we use this result together with some known results that we recall as a tool for extracting new sufficient conditions for the inverse eigenvalue problem for doubly stochastic matrices

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

on the universal realizability problem abstract we say that a list 1 2 n of complex numbers is realizable if it is the spectrum of an entrywise nonnegative matrix a we say that is universally realizable ur if is realizable for each possible jordan canonical form allowed by the universal realizability problem of spectra urp has been solved for certain kind of real and complex spectra mainly spectra in the left half plane in this paper we look for extending the kind of spectra that are ur by given new sufficient conditions for the urp to have a solution in particular we show that some realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem are realizability criteria for the urp as well

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

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

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

small polynomial matrix presentations of nonnegative matrices we investigate the use of polynomial matrices to give efficient presentations of nonnegative matrices exhibiting prescribed spectral and algebraic behavior

perturbations on constructible lists preserving realizability in the niep and questions of monov abstract in 5 the authors showed that if 1 2 2 4 n is realizable where 1 is the perron eigenvalue and 2 is non real then so too is 1 4 t 2 t 2 t 4 n they asked if 4 can be replaced by 1 or 2 or what is the least possible multiple c 1 of t in order for this perturbation to be realizable in 2 the authors showed that for n 4 one can find certain spectra for which the result holds when c 1 provided t is small in this paper we show that c 2 is best possible and we construct a realizing matrix for c 2 when t is sufficiently small we also address some questions of monov concerning the realizability of the derivative of a realizable polynomial and if such a polynomial must have positive power sums

on positive matrices abstract we relate several results on positive matrices due to soittola 1976 handelman 1981 1987 and lind 1984 giving conditions for the existence of some positive matrix to represent a linear map the result of soittola concerns finite automata and those of handelman and lind concern symbolic dynamics

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

growth type invariants for ℤ d subshifts of finite type and arithmetical classes of real numbers we discuss some numerical invariants of multidimensional shifts of finite type sfts which are associated with the growth rates of the number of admissible finite configurations extending an unpublished example of tsirelson a strange two dimensional symbolic system 1992 we show that growth complexities of the form exp n o 1 are possible for non integer s in terminology of de carvalho port math 54 1 19 40 1997 such subshifts have entropy dimension the class of possible s are identified in terms of arithmetical classes of real numbers of weihrauch and zheng math log q 47 1 51 65 2001

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

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

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 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

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

strong shift equivalence and the generalized spectral conjecture for nonnegative matrices abstract given matrices a and b shift equivalent over a dense subring r of r with a primitive we show that b is strong shift equivalent over r to a primitive matrix this result shows that the weak form of the generalized spectral conjecture for primitive matrices implies the strong form the foundation of this work is the recent result that for any ring r the group nk 1 r of algebraic k theory classifies the refinement of shift equivalence by strong shift equivalence for matrices over r

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

the real nonnegative inverse eigenvalue problem is np hard abstract a list of complex numbers is realizable if it is the spectrum of a nonnegative matrix in 1949 suleimanova posed the nonnegative inverse eigenvalue problem niep the problem of determining which lists of complex numbers are realizable the version for reals of the niep rniep asks for realizable lists of real numbers in the literature there are many sufficient conditions or criteria for lists of real numbers to be realizable we will present an unified account of these criteria then we will see that the decision problem associated to the rniep is np hard and we will study the complexity for the decision problems associated to known criteria

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

the spectral approximation of linear operators with applications to the computation of eigenelements of differential and integral operators we are concerned with the numerical solution of the eigenvalue problem t varphi lambda varphi varphi ne 0 where t is a linear operator in a bana ch space t may represent a bounded integral operator or a closed differential operator with bounded inverse the linear operator t and its approximation t n are defined in the same space perturbation theory is then a suitable framework for our problem we present in the first part a systematic study of the various notions of convergence t n to t which imply the convergence of the eigenvalues with preservation of the multiplicities the results are then applied to practical methods for integral and differential operators in the second part we present convergence rates analytic perturbation theory is used to refine on the computed eigenelements of an integral operator and to produce localization results on the eigenelements finally superconvergence results are discussed both for integral and differential operators

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

the niep for four dimensional leslie and doubly stochastic matrices with zero trace from the coefficients of the characteristic polynomial abstract the constraints on the coefficients of the characteristic polynomials of four dimensional doubly stochastic matrices with zero trace and leslie stochastic matrices with zero trace are determined then on the basis of these constraints the regions of the complex plane consisting of those points which can serve as characteristic roots of four dimensional doubly stochastic matrices with zero trace and leslie stochastic matrices with zero trace are characterized

the inverse spectrum problem for positive generalized stochastic matrices abstract we consider the inverse spectrum problem for nonnegative matrices in particular we derive sufficient conditions for the existence of nonnegative and positive generalized stochastic and generalized doubly stochastic matrices with complex and real prescribed spectrum

the inverse eigenvalue problem for nonnegative matrices abstract let a be a nonnegative matrix with spectrum 1 2 m and b be a nonnegative matrix with spectrum 1 2 n where 1 is the perron eigenvalue of b furthermore let a maximal diagonal element of a be greater than or equal to 1 in the article we construct a nonnegative matrix c with spectrum 1 2 m 2 n this construction enables us to obtain several results on how to determine new realizable lists from known realizable lists

negativity compensation in the nonnegative inverse eigenvalue problem abstract if a set of complex numbers can be partitioned as 1 s in such a way that each i is realized as the spectrum of a nonnegative matrix say a i then is trivially realized as the spectrum of the nonnegative matrix a a i in linear algebra appl 369 2003 169 it was shown that in some cases a real set can be realized even if some of the i are not realizable themselves here we systematize and extend these results in particular allowing the sets to be complex the leading idea is that one can associate to any nonrealizable set a certain negativity n and to any realizable set a certain positivity m then under appropriate conditions if m n we can conclude that is the spectrum of a nonnegative matrix additionally we prove a complex generalization of suleimanova s theorem

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

a uniform type inverse spectrum theorem for entrywise nonnegative matrices abstract influenced by earlier results of fiedler m fiedler eigenvalues of nonnegative symmetric matrices linear algebra appl 9 1974 119 142 wuven guo wuven eigenvalues of nonnegative matrices linear algebra appl 266 1997 261 270 established interesting theorems and posed several natural questions in the inverse spectrum problem for entrywise nonnegative matrices we answer some of these of a uniform character obtaining the sharpest possible results

logarithmic voltage anomalies in random resistor networks the authors investigate the behaviour of the maximum voltage drop across the bonds in a random resistor network above the percolation threshold on the basis of numerical simulations on randomly diluted l l square lattice networks we find that the average value of the maximum voltage vmax p l exhibits a peak as a function of the bond concentration p which is located above the percolation threshold this peak value appears to grow logarithmically with l while the location of the peak appears to approach the percolation threshold very slowly as l increases to help understand these results they introduce the bubble model a quasi one dimensional structure in which the system length varies exponentially in the system width this model is exactly soluble and a rather good description of percolation in systems of greater than one dimension can be obtained moreover within the bubble model the peak value of the maximum voltage increases as in l while the peak location approaches the percolation threshold as in in l in l in good agreement with our numerical results

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

universal realizability of spectra with two positive eigenvalues abstract a list of eigenvalues is said to be realizable if it is the spectrum of a nonnegative matrix diagonalizably realizable dr if it is the spectrum of a diagonalizable nonnegative matrix and universally realizable ur if there is a nonnegative matrix with this spectrum and any possible jordan canonical form allowed by the spectrum the nonnegative inverse eigenvalue problem niep asks which lists are realizable it is known that there exist spectra that are realizable but not dr we raise the question of whether dr implies ur this is known in a few cases including n 4 nonnegative spectra and suleimanova spectra we add some new classes of spectra that are ur

the nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial ebl digraphs the nonnegative inverse eigenvalue problem niep is given a family of complex numbers 1 n find necessary and sufficient conditions for the existence of a nonnegative matrix a of order n with spectrum loewy and london r loewy d london a note on the inverse eigenvalue problems for nonnegative matrices linear and multilinear algebra 6 1978 83 90 resolved it for n 3 and for n 4 when the spectrum is real in our way of handling the niep we focus our attention on the coefficients of the characteristic polynomial of a thus the niep that we consider is given k1 k2 kn real numbers find necessary and sufficient conditions for the existence of a nonnegative matrix a of order n with characteristic polynomial xn k1xn 1 k2xn 2 kn the coefficients of the characteristic polynomial are closely related to the cyclic structure of the weighted digraph with adjacency matrix a we introduce a special type of digraph structure that we shall call ebl in which this relation is specially simple we give some results that show the interest of ebl structures we completely solve the niep from the coefficients of the characteristic polynomial for n 4 we also solve a special case of the niep for n 2p 1 with k1 kp 1 0 and p 2

existence and construction of nonnegative matrices with prescribed spectrum abstract we consider the following inverse spectrum problem for nonnegative matrices given a set of real numbers 1 2 n find necessary and sufficient conditions for the existence of an n n nonnegative matrix a with spectrum in particular by the use of a relevant theorem of brauer we obtain new simple sufficient conditions for the problem to have a solution moreover we can always construct a solution matrix which is nonnegative generalized stochastic

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

approximation of the spectrum of closed operators the determination of normal modes of a rotating basin note univ michigan dept math ann arbor mi 48109 ecole polytech ctr math appl f 91128 palaiseau france descloux j ecole prat hautes etud dept math f 75231 paris 05 france isi document delivery no le068times cited 3cited reference count 17 reference asn article 1981 004doi 10 2307 2007731 record created on 2006 08 24 modified on 2017 05 12

existence and construction of nonnegative matrices with complex spectrum abstract the following inverse spectrum problem for nonnegative matrices is considered given a set of complex numbers 1 2 n find necessary and sufficient conditions for the existence of an n n nonnegative matrix a with spectrum our work is motivated by a relevant theoretical result of guo wuwen linear algebra appl 266 1997 261 theorem 2 1 there exists a real parameter 0 max 2 j n j such that the problem has a solution if and only if 1 0 in particular we discuss how to compute 0 and the solution matrix a for certain class of matrices a sufficient condition for the problem to have a solution is also derived

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

temporal development of invasion percolation the authors consider the dynamics of invasion percolation for different models this process exhibits scaling behaviours which in the simplest case can be related to some critical static properties of usual percolation in particular the concept of bursts is shown to be of fundamental importance the critical exponents they derive are checked numerically on a case of invasion percolation where growth is restricted to the hull of the existing cluster

on nonnegative matrices similar to positive matrices abstract let nn denote the set of those 1 2 n c n such that there exists a nonnegative matrix with perron root equal to one and spectrum 1 2 n we prove that nn is star shaped with respect to 1 0 0 and that 1 2 n n n is on the boundary of nn if and only if 1 2 n is not the spectrum of any positive matrix as a consequence attention is given to the problem of determining which nonnegative matrices are similar to positive ones more precisely we address the question of which pattern matrices p satisfy that any nonnegative matrix with pattern p is similar to a positive matrix some partial results are obtained among them that any irreducible nonnegative matrix with a positive line is similar to a positive matrix which allow us to give a complete solution to the case of 3 by 3 matrices

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

sampling eigenvalues in hardy spaces in this work we extend the sampling method to compute eigenvalues of singular non self adjoint sturm liouville problems in the presence of a continuous spectrum we first show that the characteristic function whose zeros are the eigenvalues belongs to a hardy space and then develop a new sampling formula for its reconstruction we estimate the truncation error obtain computable error bounds and test the method with a few numerical experiments

on relationships between several classes of z matrices m matrices and nonnegative matrices abstract in this paper we establish relationships between several classes of well known matrices and some new classes of matrices we consider matrices whose powers are irreducible z matrices or m matrices these matrices are called zma matrices or mma matrices we establish a relationship between theses classe of matrices and the class of soules matrices moreover we prove that symmetric zma matrices and mma matrices are diagonally congruent to matrices that we call z sums and m sums conversely z sums and m sums are diagonally congruent to zma matrices and mma matrices this relationship exploits the structure of zma matrices and mma matrices and several well known properties of zma matrices follow immediately z sums and m sums are closely related to strictly ultrametric matrices sums we also characterize inverse zma matrices we prove that symmetric inverse zma matrices are diagonally congruent to shifted sums and that shifted sums are diagonally congruent to symmetric inverse zma matrices

extreme nonnegative matrices abstract an entrywise nonnegative n n matrix a is extreme if its spectrum 1 n has the property that for all 0 1 n is not the spectrum of a nonnegative matrix it is proved that if a is an extreme nonnegative matrix then there exists a nonzero nonnegative matrix y such that ay ya and at y 0 where is the hadamard or entrywise product and t denotes transpose

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

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

a characterization of trace zero symmetric nonnegative 5 5 matrices abstract the problem of determining necessary and sufficient conditions for a set of real numbers to be the eigenvalues of a symmetric nonnegative matrix is called the symmetric nonnegative inverse eigenvalue problem sniep in this paper we solve sniep in the case of trace zero symmetric nonnegative 5 5 matrices

an inverse eigenvalue problem for nonnegative matrices abstract this paper gives a solution to an inverse eigenvalue problem of nonnegative matrices which was proposed by miroslav fiedler

a characterization of trace zero bisymmetric nonnegative 5 5 matrices abstract let 1 2 3 4 5 1 be real numbers such that i 1 5 i 0 in 16 o spector proved that a necessary and sufficient condition for 1 2 3 4 5 to be the eigenvalues of a traceless symmetric nonnegative 5 5 matrix is 2 5 0 and i 1 5 i 3 0 in this article we show that this condition is also a necessary and sufficient condition for 1 2 3 4 5 to be the spectrum of a traceless bisymmetric nonnegative 5 5 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

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

on the inverse eigenvalue problem for nonnegative matrices of order two to five abstract in this paper for a given set of numbers with special conditions we construct a nonnegative matrix such that the given set is its spectrum

nonnegative realizability with jordan structure abstract a general method is given for merging blocks in the jordan canonical form of a nonnegative matrix as a consequence results more general than any prior ones are given for the universal realizability of spectra that is spectra which are realizable by a nonnegative matrix for each possible jordan canonical form allowed by the spectrum in particular we generalize a classical result due to minc regarding positive diagonalizable matrices for example any spectrum that is diagonalizably realizable by a nonnegative matrix with mostly positive off diagonal entries is universally realizable

nonnegative matrices with prescribed spectrum and elementary divisors abstract in this paper we give new sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors which drastically improve and contain some of the previous known conditions we also show how to perturb complex eigenvalues of a nonnegative matrix while keeping its nonnegativity these results allow us under certain conditions to easily decide if a given list is realizable with prescribed elementary divisors

a note on the construction of a positive oscillatory matrix with a prescribed spectrum a necessary and sufficient condition for an n tuple of real numbers 1 2 n to be the spectrum of an oscillatory matrix is that 1 2 n 0 some methods of constructing a positive symmetric oscillatory matrix with spectrum 1 2 n are suggested for theorems concerning oscillatory matrices in this work we present a new procedure of constructing such a matrix which is numerically stable and appears to be less expensive

the symmetric nonnegative inverse eigenvalue problem for 5 5 matrices abstract the symmetric nonnegative inverse eigenvalue problem sniep asks when a list 1 2 n of n real numbers is the spectrum of an n n symmetric nonnegative matrix this problem is completely solved only for n 4 our main goal here is to contribute to the solution of sniep for n 5 we also give a sufficient condition for a list to be realized as the spectrum of a symmetric positive matrix

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

power series with positive coefficients arising from the characteristic polynomials of positive matrices let a be an n times n entrywise positive matrix and let f t det i t a we prove the surprising result that there always exists a positive integer n such that the formal power series expansion of 1 f t 1 n around t 0 has positive coefficients

the spectra of nonnegative integer matrices via formal power series an old problem in matrix theory is to determine the n tuples of complex numbers which can occur as the spectrum of a matrix with nonnegative entries see bp94 chapter 4 or min88 chapter vii authors have studied the case where the ntuple is comprised of real numbers bor95 cia68 fri78 kel71 per53 sal72 sou83 sul49 the case where the matrices under consideration are symmetric fie74 jll96 and the general problem joh81 lm99 ll79 rea94 rea96 wuw97 various necessary conditions and sufficient conditions have been provided but a complete characterization is known for real n tuples only for n 4 kel71 sul49 and for complex n tuples only for n 3 ll79 motivated by symbolic dynamics boyle and handelman refocused attention on the nonzero part of the spectrum by making the following spectral conjecture bh91 bh93 see also boy93 8 and lm95 chapter 11 below a matrix a is primitive if all entries of a are nonnegative and for some n all entries of a are strictly positive also

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

galerkin eigenvector approximations how close are galerkin eigenvectors to the best approximation available out of the trial subspace under a variety of conditions the galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors both orthogonal galerkin and petrov galerkin methods are considered here with a special emphasis on nonselfadjoint problems thus extending earlier studies by chatelin babuska and osborn and knyazev consequences for the numerical treatment of elliptic pdes discretized either with finite element methods or with spectral methods are discussed new lower bounds to the sep of a pair of operators are developed as well

on universal realizability of spectra abstract a list 1 2 n of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix the list is said to be universally realizable ur if it is the spectrum of a nonnegative matrix for each possible jordan canonical form allowed by it is well known that an n n nonnegative matrix a is co spectral to a nonnegative matrix b with constant row sums in this paper we extend the co spectrality between a and b to a similarity between a and b when the perron eigenvalue is simple we also show that if 0 and 1 2 n is ur then 1 2 n is also ur we give counter examples for the cases 1 2 n is ur implies 1 2 3 n is ur and 1 2 are ur implies 1 2 is ur

spectra of principal submatrices of nonnegative matrices abstract let a be an n n nonnegative matrix with the spectrum 1 2 n and let a 1 be an m m principal submatrix of a with the spectrum 1 2 m in this paper we present some cases where the realizability of 1 2 m 1 2 s implies the realizability of 1 2 n 1 2 s and consider the question whether this holds in general in particular we show that the list 1 2 n 1 2 m is realizable

on continuity of guo wuwen function abstract we show that the functions g and g s introduced by guo wuwen in 4 are continuous and semialgebraic we use this fact to prove that the set n n of ordered n tuples of real numbers realizable by nonnegative matrices is a closed set

conductance and resistance jumps in finite size random resistor networks when bonds are removed one by one and at random from a finite size resistor network the conductance or resistance does not change continuously but rather a sequence of conductance resistance jumps for various sizes occurs the larger jumps arise from those bonds which carry a relatively large current just before they are cut the authors report on numerical simulations of these jumps of a random resistor network on the square lattice they also give a scaling argument to account for this phenomenon which yields the number of conductance jumps larger than delta g scaling as delta g lambda g with lambda g d nu d nu t where t is the conductivity exponent and nu is the percolation correlation length exponent equivalently the number of resistance jumps greater than delta r scales as delta r lambda r with lambda r d nu d nu t these predictions account for the data on a qualitative level only however and they discuss some possible mechanisms for the quantitative discrepancies

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

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

on the spectra of some g circulant matrices and applications to nonnegative inverse eigenvalue problem abstract a g circulant matrix a is defined as a matrix of order n where the elements of each row of a are identical to those of the previous row but are moved g positions to the right and wrapped around using number theory certain spectra of g circulant real matrices are given explicitly the obtained results are applied to nonnegative inverse eigenvalue problem to construct nonnegative g circulant matrices with given appropriated spectrum additionally some g circulant matrices are reconstructed from its main diagonal entries