on newton s method and halley s method for the principal pth root of a matrix if a is a matrix with no negative real eigenvalues and all zero eigenvalues of a are semisimple the principal pth root of a can be computed by newton s method or halley s method with a preprocessing procedure if necessary we prove a new convergence result for newton s method and discover an interesting property of newton s method and halley s method in terms of series expansions we explain how the convergence of newton s method and halley s method can be improved when the eigenvalues of a are known or when a is a singular matrix we also prove new results on pth roots of m matrices and h matrices and consider the application of newton s method and halley s method to find the principal pth roots of these special matrices ams classification 65f30 41a58 15a48

perron frobenius theorem for matrices with some negative entries abstract we extend the classical perron frobenius theorem to matrices with some negative entries we study the cone of matrices that has the matrix of 1 s ee t as the central ray and such that any matrix inside the cone possesses a perron frobenius eigenpair we also find the maximum angle of any matrix with ee t that guarantees this property

infinitely divisible nonnegative matrices m matrices and the embedding problem for finite state stationary markov chains abstract this paper explicitly details the relation between m matrices nonnegative roots of nonnegative matrices and the embedding problem for finite state stationary markov chains the set of nonsingular nonnegative matrices with arbitrary nonnegative roots is shown to be the closure of the set of matrices with matrix roots in im the methods presented here employ nothing beyond basic matrix analysis however it answers a question regarding m matrices posed over 30 years ago and as an application a new characterization of the set of all embeddable stochastic matrices is obtained as a corollary

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

stochastic non convex envelopment of data applying isotonic regression to frontier estimation abstract isotonic nonparametric least squares inls is a regression method for estimating a monotonic function by fitting a step function to data in the literature of frontier estimation the free disposal hull fdh method is similarly based on the minimal assumption of monotonicity in this paper we link these two separately developed nonparametric methods by showing that fdh is a sign constrained variant of inls we also discuss the connections to related methods such as data envelopment analysis dea and convex nonparametric least squares cnls further we examine alternative ways of applying isotonic regression to frontier estimation analogous to corrected and modified ordinary least squares cols mols methods known in the parametric stream of frontier literature we find that inls is a useful extension to the toolbox of frontier estimation both in the deterministic and stochastic settings in the absence of noise the corrected inls cinls has a higher discriminating power than fdh in the case of noisy data we propose to apply the method of non convex stochastic envelopment of data non convex stoned which disentangles inefficiency from noise based on the skewness of the inls residuals the proposed methods are illustrated by means of simulated examples

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

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

an explicit a posteriori error estimations for the cell functional minimization scheme of elliptic problems abstract this paper presents an explicit form of a posteriori error estimator for the cell functional minimization scheme of elliptic problems it is shown that this estimator is reliable with respect to an energy type error numerical experiments are conducted to demonstrate the performance of the error estimator in the adaptive mesh refinement algorithm

asymptotic properties of powers of nonnegative matrices with applications abstract let a be a finite square reducible nonnegative matrix the main theorem of the paper gives first order approximations of f n a for certain sequences f n of analytic functions in particular the theorem holds when f n a a n the theorem is applied to study the local behavior of large powers of a nonnegative matrix to study the limiting output vectors of a nonnegative multiplicative process and to characterize the nonnegative eigenvectors of a nonnegative matrix finally an application to absorbing markov chains is given

paths in directed graphs and spectral properties of matrices abstract the study of the relationship between graph theoretic properties and spectral properties of matrices has been of interest in the past eighty years for about seventy years research focused on nonnegative matrices but in the past decade the investigation has been extended to general matrices over an arbitrary field several papers have been devoted to the relationship between the structure of the jordan blocks associated with the eigenvalue 0 of a matrix and the digraph of the matrix this problem appears to be strongly linked to a graph theoretic study of paths in digraphs this paper reviews the development of the above mentioned studies it proceeds simultaneously with the graph theoretic and the matrix theoretic problems showing the links at each stage

the combinatorial structure of generalized eigenspaces from nonnegative matrices to general matrices abstract the perron frobenius spectral theory of nonnegative matrices motivated an intensive study of the relationship between graph theoretic properties and spectral properties of matrices while for about seventy years research focused on nonnegative matrices in the past fifteen years the study has been extended to general matrices over an arbitrary field one of the major original problems in this context is determining the relations between the matrix analytic height characteristic of a matrix and the graph theoretic level characteristic in this article the history of this problem is reviewed from its introduction for nonnegative matrices through its complete solution for nonnegative matrices to the solution of the generalized version of the problem for general matrices

predictable dynamics of opinion forming for networks with antagonistic interactions for communities of agents which are not necessarily cooperating distributed processes of opinion forming are naturally represented by signed graphs with positive edges representing friendly and cooperative interactions and negative edges the corresponding antagonistic counterpart unlike for nonnegative graphs the outcome of a dynamical system evolving on a signed graph is not obvious and it is in general difficult to characterize even when the dynamics are linear in this paper we identify a significant class of signed graphs for which the linear dynamics are however predictable and show many analogies with positive dynamical systems these cases correspond to adjacency matrices that are eventually positive for which the perron frobenius property still holds and implies the existence of an invariant cone contained inside the positive orthant as examples of applications we determine cases in which it is possible to anticipate or impose unanimity of opinion in decision voting processes even in presence of stubborn agents and show how it is possible to extend the pagerank algorithm to include negative links

eventually and asymptotically positive semigroups on banach lattices abstract we develop a theory of eventually positive c 0 semigroups on banach lattices that is of semigroups for which for every positive initial value the solution of the corresponding cauchy problem becomes positive for large times we give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators complementing existing results on spaces of continuous functions this enables us to treat a range of new examples including the square of the laplacian with dirichlet boundary conditions the bi laplacian on l p spaces the dirichlet to neumann operator on l 2 and the laplacian with non local boundary conditions on l 2 within the one unified theory we also introduce and analyse a weaker notion of eventual positivity which we call asymptotic positivity where trajectories associated with positive initial data converge to the positive cone in the banach lattice as t this allows us to discuss further examples which do not fall within the above mentioned framework among them a network flow with non positive mass transition and a certain delay differential equation

matrix roots of eventually positive matrices abstract eventually positive matrices are real matrices whose powers become and remain strictly positive as such eventually positive matrices are a fortiori matrix roots of positive matrices which motivates us to study the matrix roots of primitive matrices using classical matrix function theory and perron frobenius theory we characterize classify and describe in terms of the real jordan canonical form the p th roots of eventually positive matrices

sensitivity analysis of efficiency scores how to bootstrap in nonparametric frontier models efficiency scores of production units are generally measured relative to an estimated pro duction frontier nonparametric estimators dea fdh are based on a finite sample of observed production units the bootstrap is one easy way to analyze the sensitivity of efficiency scores relative to the sampling variations of the estimated frontier the main point in order to validate the bootstrap is to define a reasonable data generating process in this complex framework and to propose a reasonable estimator of it this paper provides a general methodology of bootstrapping in nonparametric frontier models some adapted methods are illustrated in analyzing the bootstrap sampling variations of input efficiency measures of electricity plants

on the perron frobenius theory for complex matrices abstract we extend here the perron frobenius theory of nonnegative matrices to certain complex matrices following the generalization of the perron frobenius theory to matrices that have some negative entries given by noutsos 14 we introduce here two types of extensions of the perron frobenius theory to complex matrices we present and prove here some sufficient conditions and some necessary and sufficient conditions for a complex matrix to have a perron frobenius eigenpair we apply this theory by introducing perron frobenius splittings as well as complex perron frobenius splittings for the solution of complex linear systems a x b by classical iterative methods perron frobenius splittings constitute an extension of the well known regular splittings weak regular splittings and nonnegative splittings convergence and comparison properties are also given and proved

inverses of m type matrices created with irreducible eventually nonnegative matrices abstract an m matrix is a matrix that can be expressed as i p where p is entry wise nonnegative and p it is well known that the inverse of a nonsingular irreducible m matrix is positive in this paper matrices of the form i p where p is an irreducible eventually nonnegative matrix and p are studied it is shown that if index 0 p 1 then there exists a positive number such that if p then the inverse of i p is positive

computing short interval transition matrices of a discrete time markov chain from partially observed data markov chains constitute a common way of modelling the progression of a chronic disease through various severity states for these models a transition matrix with the probabilities of moving from one state to another for a specific time interval is usually estimated from cohort data quite often however the cohort is observed at specific times with intervals that may be greater than the interval of interest the transition matrix computed then needs to be decomposed in order to estimate the desired interval transition matrix suited to the model although simple to implement this method of matrix decomposition can yet result in an invalid short interval transition matrix with negative or complex entries in this paper we present a method for computing short interval transition matrices that is based on regularization techniques our method operates separately on each row of the invalid short interval transition matrix aiming to minimize an appropriate distance measure we test our method on various matrix structures and sizes and evaluate its performance on a real life transition model for hiv infected individuals

testing procedures for detection of linear dependencies in efficiency models the validity of many efficiency measurement methods rely upon the assumption that variables such as input quantities and output mixes are independent of or uncorrelated with technical efficiency however few studies have attempted to test these assumptions in a recent paper wilson 2003 investigates a number of independence tests and finds that they have poor size properties and low power in moderate sample sizes in this study we discuss the implications of these assumptions in three situations i bootstrapping non parametric efficiency models ii estimating stochastic frontier models and iii obtaining aggregate measures of industry efficiency we propose a semi parametric hausmann type asymptotic test for linear independence uncorrelation and use a monte carlo experiment to show that it has good size and power properties in finite samples we also describe how the test can be generalized in order to detect higher order dependencies such as heteroscedasticity so that the test can be used to test for full independence when the efficiency distribution has a finite number of moments finally an empirical illustration is provided using data on us electric power generation

a posteriori error estimate and adaptive mesh refinement for the cell centered finite volume method for elliptic boundary value problems we extend a result of nicaise siam j numer anal 43 2005 pp 1481 1503 for the a posteriori error estimation of the cell centered finite volume method for the numerical solution of elliptic problems having computed the piecewise constant finite volume solution u h we compute a morley type interpolant mathcal i u h for the exact solution u the energy error norm nabla mathcal t u mathcal i u h l 2 can be controlled efficiently and reliably by a residual based a posteriori error estimator eta the local contributions of eta are used to steer an adaptive mesh refining algorithm a model example serves the laplace equation in two dimensions with mixed dirichlet neumann boundary conditions

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

on complex power nonnegative matrices abstract power nonnegative matrices are defined as complex matrices having at least one nonnegative integer power we exploit the possibility of deriving a perron frobenius like theory for these matrices obtaining three main results and drawing several consequences we study in particular the relationships with the set of matrices having eventually nonnegative powers the inverse of m type matrices and the set of matrices whose columns rows sum up to one

modifying the inertia of matrices arising in optimization applications in constrained optimization and other areas produce symmetric matrices with a natural block 2 2 structure an optimality condition leads to the problem of perturbing the 1 1 block of the matrix to achieve a specific inertia we derive a perturbation of minimal norm for any unitarily invariant norm that increases the number of nonnegative eigenvalues by a given amount and we show how it can be computed efficiently given a factorization of the original matrix we also consider an alternative way to satisfy the optimality condition based on a projection approach theoretical tools developed here include an extension of ostrowski s theorem on congruences and some lemmas on inertias of block 2 2 symmetric matrices

matrix roots of imprimitive irreducible nonnegative matrices abstract using matrix function theory perron frobenius theory combinatorial matrix theory and elementary number theory we characterize classify and describe in terms of the jordan canonical form the matrix p th roots of imprimitive irreducible nonnegative matrices preliminary results concerning the matrix roots of reducible matrices are provided as well

a posteriori error estimations of some cell centered finite volume methods this paper presents the natural framework to residual based a posteriori error estimation of some cell centered finite volume methods for the laplace equation in r d d 2 or 3 for that purpose we associate with the finite volume solution a reconstructed approximation which is a kind of morley interpolant the error is then the difference between the exact solution and this morley interpolant the residual error estimator is based on the jump of normal and tangential derivatives of the morley interpolant we then prove the equivalence between the discrete h1 seminorm of the error and the residual error estimator numerical tests confirm our theoretical results

comparisons of regular splittings of matrices in this article new comparison theorems for regular splittings of matrices are derived in so doing the initial results of varga in 1960 on regular splittings of matrices and the subsequent unpublished results of wo nicki in 1973 on regular splittings of matrices will be seen to be special cases of these new comparison theorems

minimal wave speed for a class of non cooperative reaction diffusion systems of three equations abstract in this paper we study the traveling wave solutions and minimal wave speed for a class of non cooperative reaction diffusion systems consisting of three equations based on the eigenvalues a pair of upper lower solutions connecting only the invasion free equilibrium are constructed and the schauder s fixed point theorem is applied to show the existence of traveling semi fronts for an auxiliary system then the existence of traveling semi fronts of original system is obtained by limit arguments the traveling semi fronts are proved to connect another equilibrium if natural birth and death rates are not considered and to be persistent if these rates are incorporated then non existence of bounded traveling semi fronts is obtained by two sided laplace transform then the above results are applied to some disease transmission models and a predator prey model

a nonconforming a posteriori estimator for the coupling of cell centered finite volume and boundary element methods the coupling of the cell centered finite volume method and the boundary element method is an interesting approach to solve elliptic problems on an unbounded domain where local flux conservation is important based on the piecewise constant interior finite volume solution we define a morley type interpolant built on a nonconforming finite element together with the cauchy data of the exterior boundary element solution this allows us to define a residual based a posteriori error estimator with respect to an energy norm we prove reliability and efficiency of this estimator and use its local contributions to steer an adaptive mesh refining algorithm in two examples we illustrate the effectiveness of the new adaptive coupling method and compare it with the coupling approach with a conforming morley interpolant

eventually nonnegative matrices are similar to seminonnegative matrices in this paper it is shown that the necessary and sufficient conditions on the jordan form of a seminonnegative matrix identified by zaslavsky and mcdonald are in fact necessary and sufficient conditions on the jordan form of every eventually nonnegative matrix thus every eventually nonnegative matrix is similar to a seminonnegative matrix in linear algebra appl 372 2003 253 they show that several of the combinatorial properties of reducible nonnegative matrices carry over to reducible seminonnegative matrices in this paper it is shown that a property on the index of cyclicity of an irreducible nonnegative matrix carries over to the seminonnegative matrices

nonnegative and eventually positive matrices abstract we characterize those eventually positive matrices a such that the sum of a and every nonnegative matrix remains eventually positive for n 3 we show that for every eventually positive matrix a m n r there exists an eventually positive matrix b such that a b is not eventually positive

ray pattern matrices requiring perron frobenius properties abstract a ray pattern matrix is a matrix whose nonzero entries are all unimodular an n n ray pattern matrix a is said to require the weak perron frobenius property if for any n n entrywise positive matrix k the spectral radius k a is one of the eigenvalues of k a to require the strong perron frobenius property if k a is an algebraically simple eigenvalue of k a and the left and right eigenvectors associated to k a are strictly nonzero in this paper necessary and sufficient conditions for ray pattern matrices requiring the weak strong perron frobenius property are given

a schur method for the square root of a matrix abstract a fast and stable method for computing the square root x of a given matrix a x 2 a is developed the method is based on the schur factorization a qsq h and uses a fast recursion to compute the upper triangular square root of s it is shown that if x 2 a is not large then the computed square root is the exact square root of a matrix close to a the method is extended for computing the cube root of a matrices exist for which the square root computed by the schur method is ill conditioned but which nonetheless have well conditioned square roots an optimization approach is suggested for computing the well conditioned square roots in these cases

perron frobenius property of copositive matrices and a block copositivity criterion abstract haynsworth and hoffman proved in 1969 that the spectral radius of a symmetric copositive matrix is an eigenvalue of this matrix this note investigates conditions which guarantee that an eigenvector corresponding to this dominant eigenvalue has no negative coordinates i e whether the perron frobenius property holds also a block copositivity criterion using the schur complement is specified which may be helpful to reduce dimension in copositivity checks and which generalizes results proposed by andersson et al in 1995 and johnson and reams in 2005 apparently the latter five researchers were unaware of the more general results by the author precedingly published in 1987 and 1996 respectively

comparing nonparametric versus parametric regression fits in general there will be visible differences between a parametric and a nonparametric curve estimate it is therefore quite natural to compare these in order to decide whether the parametric model could be justified an asymptotic quantification is the distribution of the integrated squared difference between these curves we show that the standard way of bootstrapping this statistic fails we use and analyse a different form of bootstrapping for this task we call this method the wild bootstrap and apply it to fitting engel curves in expenditure data analysis

on the structure of the set of symmetric matrices with the perron frobenius property abstract in this paper we explore the geometry of the set of symmetric perron frobenius matrices whose spectral radius is an eigenvalue with a nonnegative associated eigenvector we prove that the set which is not convex is star convex with the identity matrix as the center we also describe a few aspects of the geometric structure of the set

minimal eventually positive realizations of externally positive systems it is a well known fact that externally positive linear systems may fail to have a minimal positive realization in order to investigate these cases we introduce the notion of minimal eventually positive realization for which the state update matrix becomes positive after a certain power eventually positive realizations capture the idea that in the impulse response of an externally positive system the state of a minimal realization may fail to be positive but only transiently as a consequence we show that in discrete time it is possible to use downsampling to obtain minimal positive realizations matching decimated sequences of markov coefficients of the impulse response in continuous time instead if the sampling time is chosen sufficiently long a minimal eventually positive realization leads always to a sampled realization which is minimal and positive

algebraically positive matrices abstract we introduce the concept of algebraically positive matrices and investigate some basic properties including a characterization the index of algebraic positivity and sign patterns that allow or require this property we also pose two open problems

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

the combinatorial structure of the generalized nullspace of a block triangular matrix the combinatorial structure of the generalized nullspace of a block triangular matrix with entries in an arbitrary field is studied using an extension lemma we prove the existence of a weakly preferred basis for the generalized nullspace independently we study the height of generalized nullvectors as a corollary we obtain the index theorem which provides an upper bound for the index of a general matrix in terms of the indices of its diagonal blocks we also investigate the case of equality in the index theorem

on the jordan form of an irreducible matrix with eventually nonnegative powers abstract in terms of the concept of a frobenius collection of elementary jordan blocks which we introduce we characterize the collection of elementary jordan blocks that appear in the jordan form of an irreducible m cyclic eventually nonnegative matrix whose m th power is permutationally similar to a direct sum of m eventually positive matrices

on pth roots of stochastic matrices in markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available the problem arises of determining a stochastic pth root of a stochastic matrix the given transition matrix by exploiting the theory of functions of matrices we develop results on the existence and characterization of matrix pth roots and in particular on the existence of stochastic pth roots of stochastic matrices our contributions include characterization of when a real matrix has a real pth root a classification of pth roots of a possibly singular matrix a sufficient condition for a pth root of a stochastic matrix to have unit row sums and the identification of two classes of stochastic matrices that have stochastic pth roots for all p we also delineate a wide variety of possible configurations as regards existence nature primary or nonprimary and number of stochastic roots and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix

towards a perron frobenius theory for eventually positive operators abstract this article is a contribution to the spectral theory of so called eventually positive operators i e operators t which may not be positive but whose powers t n become positive for large enough n while the spectral theory of such operators is well understood in finite dimensions the infinite dimensional case has received much less attention in the literature we show that several sensible notions of eventual positivity can be defined in the infinite dimensional setting and in contrast to the finite dimensional case those notions do not in general coincide we then prove a variety of typical perron frobenius type results we show that the spectral radius of an eventually positive operator is contained in the spectrum we give sufficient conditions for the spectral radius to be an eigenvalue admitting a positive eigenvector and we show that the peripheral spectrum of an eventually positive operator is a cyclic set under quite general assumptions all our results are formulated for operators on banach lattices and many of them do not impose any compactness assumptions on the operator

on the numerical characterization of the reachability cone for an essentially nonnegative matrix abstract given an n n real matrix a with nonnegative off diagonal entries the solution to x t ax t x 0 x 0 t 0 is x t e ta x 0 the problem of identifying the initial points x 0 for which x t becomes and remains entrywise nonnegative is considered it is known that such x 0 are exactly those vectors for which the iterates x k i ha k x 0 become and remain nonnegative where h is a positive not necessarily small parameter that depends on the diagonal entries of a in this paper this characterization of initial points is extended to a numerical test when a is irreducible if x k becomes and remains positive then so does x t if x t fails to become and remain positive then either x k becomes and remains negative or it always has a negative and a positive entry due to round off errors the latter case manifests itself numerically by x k converging with a relatively small convergence ratio to a positive or a negative vector an algorithm implementing this test is provided along with its numerical analysis and examples the reducible case is also discussed and a similar test is described

a characterization of jordan canonical forms which are similar to eventually nonnegative matrices with the properties of nonnegative matrices abstract in this paper we give necessary and sufficient conditions for a matrix in jordan canonical form to be similar to an eventually nonnegative matrix whose irreducible diagonal blocks satisfy the conditions identified by zaslavsky and tam and whose subdiagonal blocks with respect to its frobenius normal form are nonnegative these matrices are referred to as seminonnegative matrices and we show that they exhibit many of the same combinatorial spectral properties as nonnegative matrices this paper extends the work on jordan forms of irreducible eventually nonnegative matrices to the reducible case

structure preserving matrix means on the marcus minc stochastic matrices abstract in this paper we show that the set ps m of all m m positive definite stochastic matrices with diagonal entries bounded above by 1 m 1 is stable under the weighted geometric mean operation it is further shown that ps m is stable for some well known multivariable matrix means the least squares means for the riemannian trace metric and the kullback leibler divergence on the convex cone of m m positive definite matrices and alm ando li mathias and bmp bini meini poloni geometric means

the rank of powers of matrices in a block triangular form abstract a graph theoretic upper bound for the rank of powers of matrices in a block triangular form is given the index theorem for general matrices is derived as a corollary

on the convergence of schroder iteration functions for pth roots of complex numbers in this work a condition on the starting values that guarantees the convergence of the schroder iteration functions of any order to a pth root of a complex number is given convergence results are derived from the properties of the taylor series coefficients of a certain function the theory is illustrated by some computer generated plots of the basins of attraction

the peripheral spectrum of a nonnegative matrix abstract in this paper we provide a necessary and sufficient condition for a collection of jordan blocks to correspond to the peripheral spectrum of a nonnegative matrix we arrive at our condition by linking the tam schneider condition to the level sets with respect to the spectral radius of a nonnegative matrix

jordan chains of h cyclic matrices abstract arising from the classification of the matrix roots of a nonnegative imprimitive irreducible matrix we present results concerning the jordan chains of an h cyclic matrix we also present ancillary results applicable to nonnegative imprimitive irreducible matrices and demonstrate these results via examples

some comparison theorems for weak nonnegative splittings of bounded operators abstract the comparison of the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same matrix has arisen in the works of many authors in this paper we derive new comparison theorems for weak nonnegative splittings and weak splittings of bounded operators in a general banach space and rather general cones and in a hilbert space which extend some of the results obtained by wo nicki japan j indust appl math 11 1994 289 342 and marek and szyld numer math 44 1984 23 35 furthermore we present new theorems also for bounded operator which extend some results by csordas and varga numer math 44 1984 23 35 for weak nonnegative splittings of matrices

on perron frobenius property of matrices having some negative entries we extend the theory of nonnegative matrices to the matrices that have some negative entries we present and prove some properties which give us information when a matrix possesses a perron frobenius eigenpair we apply also this theory by proposing the perron frobenius splitting for the solution of the linear system ax b by classical iterative methods perron frobenius splittings constitute an extension of the well known regular splittings weak regular splittings and nonnegative splittings convergence and comparison properties are given and proved

a note on comparison theorems for nonnegative matrices in a recent paper 4 csordas and varga have unified and extended earlier theorems of varga in 10 and wo nicki in 11 on the comparison of the asymptotic rates of convergence of two iteration matrices induced by two regular splittings the main purpose of this note is to show a connection between the csordas varga paper and a paper by beauwens 1 in which a comparison theorem is developed for the asymptotic rate of convergence of two nonnegative iteration matrices induced by two splittings which are not necessarily regular monotonic norms already used in 1 play an important role in our work here

roots of completely positive maps abstract we introduce the concept of completely positive roots of completely positive maps on operator algebras we do this in different forms as asymptotic roots proper discrete roots and as continuous one parameter semigroups of roots we present structural and general existence and non existence results some special examples in settings where we understand the situation better and several challenging open problems our study is closely related to elfving s embedding problem in classical probability and the divisibility problem of quantum channels

on limits of powers of certain absolutely row stochastic matrices abstract we consider limits of powers of matrices that are absolutely row stochastic matrices a such that a is row stochastic we give graph theoretic criteria when such limits exist and if so determine their value in particular we show that for aperiodic connected matrices that satisfy opposition bipartiteness obipness a perron frobenius property holds here we call a obip also called structural balance in the literature when nodes in the graph corresponding to a can be partitioned into two groups such that within group links are non negative and across group links are non positive

on spectral properties of signed laplacians with connections to eventual positivity signed graphs have appeared in a broad variety of applications ranging from social networks to biological networks from distributed control and computation to power systems in this paper we investigate spectral properties of signed laplacians for undirected signed graphs we find conditions on the negative weights under which a signed laplacian is positive semidefinite via the kron reduction and multiport network theory for signed laplacians that are indefinite we characterize their inertias with the same framework furthermore we build connections between signed laplacians generalized m matrices and eventually exponentially positive matrices

on matrices with cyclic structure abstract the relations between several necessary conditions for a square complex matrix a to be m cyclic are examined these conditions are known to be equivalent if a is an irreducible nonnegative matrix whose index of imprimitivity is m in particular we find that if the digraph of a contains at least one cycle with nonzero signed length then the following conditions are each equivalent to the m cyclicity of a i a is diagonally similar to e 2 i m a ii all cycles in the digraph of a have signed length an integral multiple of m in the course of our investigations we lay down the groundwork of the theory of cyclically m partite or linearly partite digraphs and characterize these digraphs in terms of the signed lengths of their cycles the characterization of diagonal similarity between matrices in terms of matrix cycle products due to saunders and schneider plays a key role in our development our investigations also lead to a new illuminating conceptual proof for the second part of the frobenius theorem on an irreducible nonnegative matrix the connection between the m cyclicity of a square complex matrix and that of its associated collection of elementary jordan blocks is studied in the second half of this paper in particular for an m cyclic collection u it is found that the problem of determining all m tuples k 1 k m of positive integers for which there exists an m cyclic matrix a in the superdiagonal k 1 k m block form such that u a u is equivalent to determine all row sum vectors of 0 1 matrices that have a prescribed column sum vector and each of whose column vectors has cyclically consecutive equal components as a by product we also obtain an equivalent condition on m given square complex matrices b 1 b m so that there exist complex rectangular matrices a 1 a m that satisfy b j a j a m a 1 a j 1 for j 1 m thus completing the work of flanders for the case m 2

the influence of the marked reduced graph of a nonnegative matrix on the jordan form and on related properties a survey abstract for a nonnegative matrix p we discuss the relation of its marked reduced graph to that part of the jordan form that is associated with the perron frobenius root to the nonnegativity of the eigenvectors and generalized eigenvectors to the nonnegativity of solutions of linear equations and to the asymptotic growth of powers of the matrix results are often stated in terms of m matrices and standard results on irreducible matrices are assumed we give examples to illustrate the theorems surveyed

estimation of dynamic discrete models from time aggregated data an important component in dynamic discrete choice models and dynamic discrete games is the transition density of state variables from the current period to the next period most empirical dynamic discrete choice models identify the theoretical time interval in the behavioral model with that observed in the data set however many empirical data sets are time aggregated in this paper we show that when the time interval in the behavioral theory model differs from that in the observed data difficulties with nonparametric identification and specification arise in addition we study the properties of parametric maximum likelihood estimators and flexible semiparametric estimators of the transition density in dynamic discrete models with time aggregated data sets

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

a new conservative numerical scheme for flow problems on unstructured grids and unbounded domains abstract we consider the prototype of a flow and transport problem in two dimensions in particular in many fluid dynamics problems the boundary conditions may be unknown or the domain may be unbounded furthermore only numerical schemes which conserve mass and are stable with respect to dominating convection are appropriate to solve these problems therefore we develop a new discrete scheme to address these issues more precisely we couple the cell centered finite volume method and the boundary element method and introduce suitable transmission conditions also semi robust a posteriori estimates of residual type are developed which allow us to use an adaptive mesh refinement algorithm to treat efficiently problems which have singularities or boundary internal layers several examples illustrate the effectiveness of our new adaptive coupling method

comparison theorems for weak splittings of bounded operators comparison theorems for weak splittings of bounded operators are presented these theorems extend the classical comparison theorem for regular splittings of matrices by varga the less known result by wo nicki and the recent results for regular and weak regular splittings of matrices by neumann and plemmons elsner and lanzkron rose and szyld the hypotheses of the theorems presented here are weaker and the theorems hold for general banach spaces and rather general cones hypotheses are given which provide strict inequalities for the comparisons it is also shown that the comparison theorem by alefeld and volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators

on stein rosenberg type theorems for nonnegative and perron frobenius splittings abstract the stein rosenberg theorem is extended and generalized to the class of nonnegative splittings a m 1 n 1 m 2 n 2 as well as to the most generalized class of perron frobenius splittings two types of the stein rosenberg theorem are stated and proved for both classes these theorems allow us to obtain comparison results for the rate of convergence of the associated iterative methods specific assumptions are given under which the inequalities of the spectral radii become equalities or strict inequalities the theoretical results are confirmed by numerical examples

structure of the nth roots of a matrix abstract let k be a subfield of c we give a criterion for a nonsingular matrix a in mmk to have an nth root in mmk the number of similarity classes of nth roots of a is given for k r further we indicate which matrices a in mmk have infinitely many nth roots in mmk

nonnegative square roots of matrices abstract by the square root of a square matrix a we mean a matrix b that satisfies b 2 a in this paper we begin a study of the entrywise nonnegative square roots of nonnegative matrices adopting mainly a graph theoretic approach to start with we settle completely the question of existence and uniqueness of nonnegative square roots for 2 by 2 nonnegative matrices by the square of a digraph h denoted by h 2 we mean the digraph with the same vertex set as h such that i j is an arc if there is a vertex k such that i k and k j are both arcs in h we call a digraph h a square root of a digraph g if h 2 g it is observed that a necessary condition for a nonnegative matrix to have a nonnegative square root is that its digraph has a square root and also that a digraph g has a square root if and only if there exists a nonnegative matrix a with digraph g such that a has a nonnegative square root we consider when or whether certain kinds of digraphs including digraphs that are disjoint union of directed paths and circuits permutation digraphs or a special kind of bigraphs have square roots we also consider when certain kinds of nonnegative matrices including monomial matrices rank one matrices and nilpotent matrices with index two have nonnegative square roots a known characterization of loopless digraphs to have square roots due to f escalantge l montejano and t rojano is extended and amended to digraphs possibly with loops some natural open questions are also posed

a family of matrices the discretized brownian bridge and distance based regression abstract the investigation of a distance based regression model using a one dimensional set of equally spaced points as regressor values and x y as a distance function leads to the study of a family of matrices which is closely related to a discrete analog of the brownian bridge stochastic process we describe its eigenstructure and several properties recovering in particular well known results on tridiagonal toeplitz matrices and related topics

null controllability of discrete time positive control systems abstract the discrete time linear positive and quasilinear monotone systems are considered the set of initial points is restricted to be the cone of non negative vectors the input belongs to the cone of non positive vectors the necessary and sufficient conditions of null controllability for linear systems are obtained the conditions are sufficient for quasilinear systems

eventually positive semigroups of linear operators abstract we develop a systematic theory of eventually positive semigroups of linear operators mainly on spaces of continuous functions by eventually positive we mean that for every positive initial condition the solution to the corresponding cauchy problem is positive for large enough time characterisations of such semigroups are given by means of resolvent properties of the generator and perron frobenius type spectral conditions we apply these characterisations to prove eventual positivity of several examples of semigroups including some generated by fourth order elliptic operators and a delay differential equation we also consider eventually positive semigroups on arbitrary banach lattices and establish several results for their spectral bound which were previously only known for positive semigroups

on the sign patterns of the smallest signless laplacian eigenvector abstract let h be a connected bipartite graph whose signless laplacian matrix is q h suppose that the bipartition of h is s t and that x is the eigenvector of the smallest eigenvalue of q h it is well known that x is positive and constant on s and negative and constant on t the resilience of the sign pattern of x under addition of edges into the subgraph induced by either s or t is investigated and a number of cases in which the sign pattern of x persists are described