the perron generalized eigenspace and the spectral cone of a cone preserving map a unified treatment is offered to reprove known results on the following four highlights of the combinatorial spectral theory of nonnegative matrices or to extend or partly extend the results to the setting of a linear map preserving a polyhedral proper or proper cone the preferred basis theorem equivalent conditions for equality of the graph theoretic level characteristic and the spectral height characteristic the majorization relation between the two characteristics and the relation between the combinatorial properties of a nonnegative matrix and the positivity of the individual entries in its principal components this is achieved by employing the new concept of spectral cone of a cone preserving map and combining the cone theoretic methods developed in our previous papers on the geometric spectral theory of cone preserving maps with the algebraic analytic method introduced by hartwig neumann and rose and further exploited by neumann and schneider for nonnegative matrices

periodic solutions and almost periodic solutions of a neutral multispecies logarithmic population model a neutral multispecies logarithmic population model is proposed in this paper by using the contraction mapping principle and constructing a suitable lyapunov functional a set of easily applicable criteria are established for the existence uniqueness and global attractivity of positive periodic solution positive almost periodic solution of the model

on a representation of commuting maps by tensor products abstract it is shown that a representation by tensor products of degree n2 exists for every pair of commuting linear maps on an n dimensional vector space but in general not a representation of degree less than n2

maximization of a matrix function related to the dittert conjecture abstract let k n denote the set of all nonnegative n n matrices whose entries have sum n and let j n 1 n n n for k 1 n and for a k n with row sum s r 1 r n and column sums c 1 c n let k be defined by k a q k i r i j c j per a we propose a problem of maximizing k of which the dittert conjecture is a special case and obtain some results related to this problem many of known theorems for n are shown to hold for k k 1 n

a non classical solution to a hessian equation from cartan isoparametric cubic we show how to construct a non smooth solution to a hessian fully nonlinear second order uniformly elliptic equation using the cartan isoparametric cubic in 5 dimensions

a primer of perron frobenius theory for matrix polynomials abstract we present an extension of perron frobenius theory to the spectra and numerical ranges of perron polynomials namely matrix polynomials of the form l i m a m 1 m 1 a 1 a 0 where the coefficient matrices are entrywise nonnegative our approach relies on the companion matrix linearization first we recount the generalization of the perron frobenius theorem to perron polynomials and report some of its consequences subsequently we examine the role of l in multistep difference equations and provide a multistep version of the fundamental theorem of demography finally we extend issos results on the numerical range of nonnegative matrices to perron polynomials

principal submatrices of co order one with the biggest perron root abstract let a be an irreducible nonnegative matrix w be any of its indices and a w be the principal submatrix of co order one obtained from a by deleting the w th column and row denote by v ext a the set of indices w such that a w has the biggest perron root among all the principal submatrices of co order one of the original matrix a we prove that exactly one jordan block corresponds to the perron root a w of a w for every w v ext a if its size is strictly greater than one for some w v ext a then the original matrix a is permutationally similar to a lower hessenberg matrix with positive entries on the superdiagonal and in the left lower corner in other words the digraph d a of a has a hamiltonian circuit and its diameter is one less than its order in the opposite case for any w v ext a there is a unique path w i i 0 p going through w in d a such that 1 a w i has the biggest perron root for i 0 p 2 a w 0 has a right positive perron eigenvector 3 a w p has a left positive perron eigenvector 4 a w i has neither a left nor a right positive perron eigenvector for i 1 p 1 thus by the spectral criterion for a nonnegative matrix to be irreducible the submatrices a w 0 a w p combined inherit the property of irreducibility we also show that a w is irreducible for every w v ext a if any of the following holds 1 a is symmetric 2 every column row of a has at least two positive nondiagonal entries 3 a has at least two columns rows all of whose entries are positive if a is an irreducible tournament matrix then either a w is also irreducible for any w v ext a or there exist exactly two indices w in and w out in v ext a such that a w in and a w out are reducible in the last case any other principal submatrix of co order one is irreducible this shows that in the general case a one vertex deleted subdigraph with the biggest perron root need not have the best connectivity properties among all one vertex deleted subdigraphs of a given strongly connected digraph d

positive periodic solutions for a neutral delay lotka volterra system abstract applying a fixed point theorem of strict set contraction some new criteria are established for the existence of positive periodic solutions for a class of neutral delay lotka volterra systems x i t x i t r i t j 1 n a i j t x j t j 1 n b i j t x j t i j t j 1 n c i j t x j t i j t i 1 2 n compared to known results our main generalization is that delays of the derivatives are not assumed to be constants and our results are more easily verifiable

predecessor property full combinatorial column rank and the height characteristics of an m matrix abstract for a loopless acyclic transitive directed graph we give a new result about the relation between the t predecessor property of subsets of its levels and the property of having full combinatorial column rank for certain blocks in its adjacency matrix as a corollary we give new lower bounds for the i th height characteristic number of a singular m matrix an open question about determining the height characteristics of singular m matrices with a prescribed singular graph is also posed

on nonsingular m matrices abstract we extend to nonsingular m matrices the following result by g sierksma if s is a nonsingular irreducible m matrix and if x and y 0 satisfy sx y with x i 0 whenever y i x are positive this theorem has several corollaries dealing with bounds on solutions and their relative errors which we also generalize

the height characteristic of block triangular matrices it is shown that the height characteristic of a block triangular matrix a with square diagonal blocks majorizes the dual of the sequence of differences of maximal cardinalities of singular k paths in a certain graph which is determined by the height characteristics of the diagonal blocks and the reduced graph of a it is also shown that an equality holds in the generic case this result implies previously known results for the block triangular case as well as for the triangular case it also yields the index theorem for the general case with an equality in the generic case

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

block lu factorizations of m matrices it is well known that any nonsingular m matrix admits an lu factorization into m matrices with l and u lower and upper triangular respectively and any singular m matrix is permutation similar to an m matrix which admits an lu factorization into m matrices varga and cai establish necessary and sufficient conditions for a singular m matrix without permutation to allow an lu factorization with l nonsingular we generalize these results in two directions first we find necessary and sufficient conditions for the existence of an lu factorization of a singular m matrix where l and u are both permitted to be singular second we establish the minimal block structure that a block lu factorization of a singular m matrix can have when l and u are m matrices

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

on the distinguished eigenvalues of a cone preserving map abstract we generalize many known results on a nonnegative matrix concerning linear inequalities collatz wielandt sets and generalized eigenvectors to the setting of a matrix preserving a finite dimensional proper cone a simple cone theoretic proof is given for the nonnegative basis theorem for the algebraic eigenspace of a nonnegative matrix the result is also extended to a matrix preserving a polyhedral cone given proper cones k 1 and k 2 in different euclidean spaces a necessary and sufficient condition is also obtained for the existence of a nonzero matrix x which takes k 2 into k 1 and satisfies ax xb where a b are given matrices preserving k 1 and k 2 respectively this extends and answers a recent open question posed by hartwig

spectral properties of rational matrix functions with nonnegative realizations abstract if a b c is an entrywise nonnegative realization of a rational matrix function w i e w c a 1 b for a vanishing at infinity then r w inf r 0 w has no poles with r is a pole of w and r a spectral radius of a is an eigenvalue of a we prove that if the realization is minimal nonnegative then 1 1 r w r a 2 2 order of the pole r w of w order of the pole r a of a 1 we characterize the order of these poles in the spirit of rothblum s index theorem namely as the length of the longest chains of singular vertices in the reduced graph of a with a suitable new access relation which incorporates b and c into the familiar access relation of a

a jordan algebra approach to the cubic eiconal equation we establish a natural correspondence between the equivalence classes of cubic solutions of a cubic eiconal equation and the isomorphy classes of cubic jordan algebras

level characteristics corresponding to peripheral eigenvalues of a nonnegative matrix in this paper we give necessary and sufficient conditions for a set of jordan blocks to correspond to the peripheral spectrum of a nonnegative matrix for each eigenvalue the level characteristic with respect to the spectral radius is defined the necessary and sufficient conditions include a requirement that the level characteristic is majorized by the height characteristic an algorithm which has been implemented in matlab is given to determine when a multiset of jordan blocks corresponds to the peripheral spectrum of a nonnegative matrix the algorithm is based on the necessary and sufficient conditions given in this paper

on spectra of expansion graphs and matrix polynomials abstract an expansion graph of a directed weighted graph g0 is obtained by replacing certain edges of g0 by disjoint chains the adjacency matrix of the expansion graph is a partial linearization of a monic matrix polynomial we prove results on common properties of a monic operator polynomial and its partial linearization the graph g0 is connected if and only if each expansion graph of g0 is connected in this case we compute the index of imprimitivity of the adjacency matrix of some special expansion graphs of g0

a majorization relation between the height and the level characteristics abstract a recently proven graph theoretic lower bound for the nullity of powers of matrices in a triangular block form is used to obtain a majorization relation between the height and the level characteristics of a general matrix an application of the result to the m matrix case yields an improvement of a previously known result

two cores of a nonnegative matrix we prove that the sequence of eigencones i e cones of nonnegative eigenvectors of positive powers a k of a nonnegative square matrix a is periodic both in max algebra and in nonnegative linear algebra using an argument of pullman we also show that the minkowski sum of the eigencones of powers of a is equal to the core of a defined as the intersection of nonnegative column spans of matrix powers also in max algebra based on this we describe the set of extremal rays of the core the spectral theory of matrix powers and the theory of matrix core is developed in max algebra and in nonnegative linear algebra simultaneously wherever possible in order to unify and compare both versions of the same theory

asymptotic exponential cones of metzler matrices and their use in the solution of an algebraic problem the aim of this paper is that of investigating the asymptotic exponential cone of a single metzler matrix introduced in 23 and of defining and analysing the new concept of asymptotic exponential cone of a family of metzler matrices along a certain direction these results will provide necessary and or sufficient conditions for the solvability of an interesting algebraic problem that arises in the context of continuous time positive switched systems and specifically in the investigation of the reachability property 21 22 25

on the perron roots of principal submatrices of co order one of irreducible nonnegative matrices let a be an irreducible nonnegative matrix and a be the perron root spectral radius of a denote by min a the minimum of the perron roots of all the principal submatrices of co order one it is well known that the interval min a a does not contain any eigenvalues of a consider any principal submatrix a v of co order one whose perron root is equal to min a we show that the jordan structure of min a as an eigenvalue of a is obtained from that of the perron root of a v as follows one largest jordan block disappears and the others remain the same so if only one jordan block corresponds to the perron root of the submatrix then min a is not an eigenvalue of a by schneider s theorem this holds if and only if there is a hamiltonian chain in the singular digraph of a v in the general case the jordan structure for the perron root of the submatrix a v and therefore that for the eigenvalue min a of a can be arbitrary but if the perron root a w of a principal submatrix a w of co order one is strictly greater than min a then a w is a simple eigenvalue of a w we also obtain different representations for the generalized eigenvectors corresponding to the eigenvalues of a contained in the annulus min a a

positive periodic solution for a neutral delay competitive system in this paper we study the existence of positive periodic solutions for a class of neutral type competitive system with delay applying the continuation theorem of coincidence degree theory a general criteria on the existence of positive periodic solutions are established some the related results are generalized and improved it is shown that the arguments delays have no effect on the existence positive periodic solution of system

on positive periodic solutions of a nonautonomous neutral delay n species competitive system abstract the purpose of this paper is to investigate a general neutral delay n species competitive nonautonomous system different from gaines and mawhin s continuation theorem r e gaines j l mawhin coincidence degree and nonlinear differential equations springer verlag berlin 1977 and abstract continuation theory for k set contraction w petryshyn z yu existence theorem for periodic solutions of higher order nonlinear periodic boundary value problems nonlinear anal 6 1982 943 969 r e gaines j l mawhin lecture notes in mathematics vol 586 springer verlag berlin 1977 a new existence theorem is introduced with the help of the theorem and some techniques a set of sufficient conditions are derived for the existence of at least one strictly positive componentwise periodic solution as an application we also examine some special cases which have been studied extensively in the literature some known results are improved and generalized

a new iterative criterion for h matrices the reducible case abstract h matrices appear in various areas of science and engineering and it is of vital importance to have an algorithm to identify the h matrix character of a certain matrix a c n n recently the present authors have proposed a new iterative criterion algorithm ah to completely identify the h matrix property of an irreducible matrix the present work extends the previous algorithm to cover the reducible case as well

on commuting matrices in max algebra and in classical nonnegative algebra abstract this paper studies commuting matrices in max algebra and nonnegative linear algebra our starting point is the existence of a common eigenvector which directly leads to max analogs and nonnegative analogs of some classical results for complex matrices we also investigate frobenius normal forms of commuting matrices particularly when the perron roots of the components are distinct for the case of max algebra we show how the intersection of eigencones of commuting matrices can be described and we consider connections with boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode

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

linear equations over cones and collatz wielandt numbers let k be a proper cone in r n let a be an n x n real matrix that satisfies ak k let b be a given vector of k and let be a given positive real number the following two linear equations are considered in this paper i i n a x b x k and ii a l n x b x k we obtain several equivalent conditions for the solvability of the first equation for the second equation we give an equivalent condition for its solvability in case when b a and we also find a necessary condition when b a and also when b a sufficiently close to b a where b a denotes the local spectral radius of a at b with fixed we also consider the questions of when the set a l n k k equals 0 or k and what the face of k generated by the set is then we derive some new results about local spectral radii and collatz wielandt sets or numbers associated with a cone preserving map and extend a known characterization of m matrices among z matrices in terms of alternating sequences

periodic solution of a periodic neutral delay equation abstract sufficient conditions are obtained for the existence of a positive periodic solution of the periodic neutral delay equation t n t a t n j 1 b j t n t j j 1 n c j t t j which arise in a food limited population model

perron frobenius theory over real closed fields and fractional power series expansions abstract some of the main results of the perron frobenius theory of square nonnegative matrices over the reals are extended to matrices with elements in a real closed field we use the results to prove the existence of a fractional power series expansion for the perron frobenius eigenvalue and normalized eigenvector of real square nonnegative irreducible matrices which are obtained by perturbing a possibly reducible nonnegative matrix further we identify a system of equations and inequalities whose solution yields the coefficients of these expansions for irreducible matrices our analysis assures that any solution of this system yields a fractional power series with a positive radius of convergence

totally expanding multiplicative systems abstract a single matrix multiplicative system consists of an n n nonnegative matrix q and an n 1 semi positive vector x 0 this system is said to be totally expanding if each entry of the sequence q n x 0 n 0 1 is unbounded a multiple matrix multiplicative system replaces q by a set q d of n n nonnegative matrices where d is in product form and is said to be totally expanding if for every in d each entry of the sequence q n x 0 n 0 1 is unbounded each of these systems is shown to be totally expanding if and only if it has no degenerate coordinates and a particular set of linear inequalities has a solution these sets of linear inequalities can also be used to approximate the smallest coordinate dependent growth rate of the output of the respective system

characterizations and classifications of m matrices using generalized nullspaces several characterizations of the class of m matrices as a subclass of the class of z matrices are given these characterizations involve alternating sequences decom positions and splittings and all are related to generalized nullspaces

on the invariant faces associated with a cone preserving map for an n n nonnegative matrix p an isomorphism is obtained between the lattice of initial subsets of 1 n for p and the lattice of p invariant faces of the nonnegative orthant r motivated by this isomorphism we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone preserving map on a polyhedral cone formulated in terms of its invariant faces in particular we obtain the following extension of the famous rothblum index theorem for a nonnegative matrix if a leaves invariant a polyhedral cone k then for each distinguished eigenvalue ofa fork there is a chain of m distinct a invariant join irreducible faces of k each containing in its relative interior a generalized eigenvector of a corresponding to referred to as semi distinguished a invariant faces associated with where m is the maximal order of distinguished generalized eigenvectors of a corresponding to but there is no such chain with more than m members we introduce the important new concepts of semi distinguished a invariant faces and of spectral pairs of faces associated with a cone preserving map and obtain several properties of a cone preserving map that mostly involve these two concepts when the underlying cone is polyhedral perfect or strictly convex and or smooth or is the cone of all real polynomials of degree not exceeding n that are nonnegative on a closed interval plentiful illustrative examples are provided some open problems are posed at the end 1991 mathematics subject classification primary 15a48 secondary 47b65 47a25 46b42

on the singular graph and jordan diagram of strictly lower triangular matrices and m matrices abstract let a be a singular m matrix or a strictly lower triangular matrix the singular graph and level diagram of a are denoted by s a and s a respectively the jordan diagram of a for 0 is denoted by j a the purpose of this paper is to answer the question what are the relations of s a and s a to j a in the case ind a 2 the following questions raised by h schneider are discussed a given j a what are the possible s b for m matrices b such that j b j a b given s a what are the possible j b for m matrices b such that s b s a

on the elementary divisors of the sylvester and lyapunov maps the paper addresses the problem of computing the elementary divisors of the tensor product of linear transformations using the analysis of the tensor products of polynomial models as developed in fuhrmann and helmke 5 we use this to study the elementary divisors of the lyapunov and complementary lyapunov maps

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

solutions of z matrix equations we investigate the existence and the nature of the solutions of the matrix equation ax b where a is a z matrix and b is a nonnegative vector when x is required to he nonnegative then an existence theorem is due to carlson and victory and is re proved in this paper we apply our results to study nonnegative vectors in the range of z matrices

reachability properties of continuous time positive systems in this technical note reachability properties of continuous time positive systems are introduced and characterized specifically first it is shown that reachability and strong reachability are equivalent properties and thus the characterization of strong reachability derived in is extended to the weaker notion of reachability in the second part of the technical note essential reachability is introduced and necessary and sufficient conditions for this property to hold are provided

survival in a quasi death process we consider a markov chain in continuous time with an absorbing coffin state and a finite set s of transient states when s is irreducible the limiting distribution of the chain as t to infty conditional on survival up to time t is known to equal the unique quasi stationary distribution of the chain we address the problem of generalizing this result to a setting in which s may be reducible and obtain a complete solution if the eigenvalue with maximal real part of the generator of the sub markov chain on s has multiplicity one the result is applied to pure death processes and more generally to quasi death processes

the relation between the jordan structure of a matrix and its graph abstract it is shown that the height characteristics of a matrix a strongly majorizes the dual sequence of the sequence of differences of maximal cardinalities of nonclosable k paths in g a and that in the generic case the height characteristics is equal to that dual sequence simultaneously it is shown that the sequence of differences of minimal k th nonclosable norms of path coverings for a directed graph g is the dual of the sequence of differences of maximal cardinalities of nonclosable k paths in g these results generalize both matrix theoretical and graph theoretical known results for the triangular case

on the generalized nullspace of m matrices and z matrices a proof is given for the preferred basis theorem for the generalized nullspace of a given m matrix the theorem is then generalized and extended to the case of a z matrix

idempotence for sign pattern matrices abstract we first characterize the n by n irreducible sign pattern matrices a that are sign idempotent that is a a 2 to identify the n by n reducible sign idempotent patterns we develop the upper diagonal completion process to find the sign pattern of each off diagonal block in an upper block triangular sign pattern matrix a so that a a 2 next we analyze the completion process qualitatively and then discuss a graph theoretic interpretation of it we then formulate the first characterization of sign idempotent patterns in terms of the upper diagonal completion process finally we establish a graph theoretic characterization of sign idempotent patterns

graph theoretically determined jordan block size structure of regular matrix pencils abstract the authors investigate the sizes of jordan blocks of regular matrix pencils by means of a one to one correspondence between a matrix pencil e a and a weighted digraph g e a based on the relationship between determinantal divisors of a pencil and spanning cycle families of the associated digraph g e a the jordan block size structure is determined graph theoretically for classes of structurally equivalent matrix pencils defined by a pair of structure matrices e a the generic jordan block sizes corresponding to the characteristic roots at zero and at infinity can be obtained from the unweighted digraph g e a eigenvalues of matrices are discussed as special cases a nontrivial mechanical example illustrates the procedure

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

jordan algebras exceptional groups and bhargava composition abstract we consider an integral version of the freudenthal construction relating jordan algebras and exceptional algebraic groups we show how this construction is related to higher composition laws of m bhargava in number theory we propose an algorithmic approach to studying orbit spaces of groups underlying bhargava s higher composition laws using this method we discover two new examples of spaces sharing similar properties and indicate several more examples of spaces where such composition laws may be introduced

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

elementary divisors of tensor products and p ranks of binomial matrices abstract let f be a field of characteristic p and let a and b be two jordan blocks with nonzero eigenvalues the reduction formulas of c w norman enable us to compute the elementary divisors of a b we investigate connections between the elementary divisors of a b and binomial matrices in particular we show that norman s reduction formulas can be used to compute the p rank of certain binomial matrices

stability and stabilizability of continuous time linear compartmental switched systems in this paper we introduce continuous time linear compartmental switched systems and investigate their stability and stabilizability properties by their nature these systems are always stable necessary and sufficient conditions for asymptotic stability for arbitrary switching functions and sufficient conditions for asymptotic stability under certain dwell time conditions on the switching functions are proposed finally stabilizability is thoroughly investigated and proved to be equivalent to the existence of a hurwitz convex combination of the subsystem matrices a condition that for positive switched systems is only sufficient for stabilizability

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 collatz wielandt sets associated with a cone preserving map abstract let k be a closed pointed full cone in r n in their treatment of perron frobenius theory for a linear map a preserving k by wielandt s approach barker and schneider introduced four sets namely 1 and 1 the collatz wielandt sets associated with a we determine the greatest lower bound and the least upper bound of these sets some known results for a nonnegative matrix relating its spectral radius to its upper and lower collatz wielandt number are extended to the setting of a cone preserving map applications of our results to the nonnegativity of solutions of linear inequalities associated with a nonnegative matrix are also considered

combinatorial verification of the elementary divisors of tensor products abstract using combinatorial arguments we determine the determinantal divisors for the tensor product of two matrices thereby obtaining its elementary divisors and jordan canonical form

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

a solution of a nonlinear system arising in spectral perturbation theory of nonnegative matrices abstract let p and e be two n n complex matrices such that for sufficiently small positive e p ee is nonnegative and irreducible it is known that the spectral radius of p ee and corresponding normalized eigenvector have fractional power series expansions the goal of the paper is to develop an algorithm for computing the coefficients of these expansions under two restrictive assumptions namely that p has a single jordan block corresponding to its spectral radius and that the unique up to scalar multiples left and right eigenvectors of p corresponding to its spectral radius say v and w satisfy v t ew 0 our approach is to consider an associated countable system of nonlinear equations and solve this system recursively at each step we consider the coefficients of the expansion of the spectral radius of p ee as parameters and solve a related linear system parametrically the next coefficient of the expansion of the spectral radius is then determined from feasibility considerations for a linear system this solution method is novel and seems useful for computing coefficients of corresponding expansions when the two restrictive assumptions are relaxed also interestingly the coefficients we compute yield a preferred basis of the generalized eigenspace corresponding to the spectral radius of the unperturbed matrix p

hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs abstract for a nonnegative n n matrix a we find that there is a polynomial f x r x such that f a is a positive matrix of rank one if and only if a is irreducible furthermore we show that the lowest degree such polynomial f x with tr f a n is unique thus generalizing the well known definition of the hoffman polynomial of a strongly connected regular digraph for any irreducible nonnegative n n matrix a we are led to define its hoffman polynomial to be the polynomial f x of minimum degree satisfying that f a is positive and has rank 1 and trace n the hoffman polynomial of a strongly connected digraph is defined to be the hoffman polynomial of its adjacency matrix we collect in this paper some basic results and open problems related to the concept of hoffman polynomials

an algebraic analytic approach to nonnegative bases abstract rothblum and independently richman and schneider have used a combination of graph theoretic and algebraic approaches based on the frobenius normal form of a matrix to show that if a is an n n nonnegative matrix then the generalized eigenspace of a corresponding to its perron root a has a nonnegative basis moreover they showed that among the jordan chains corresponding to a a has a chain of maximal length consisting entirely of nonnegative generalized eigenvectors in this paper we develop an algebraic analytic approach to the proofs of these results which is based on the asymptotic behavior of the resolvent operator the approach allows us to suggest an algorithm for computing a nonnegative basis

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

on the zero pattern properties and asymptotic behavior of continuous time positive system trajectories abstract in this paper the zero pattern properties and the asymptotic evolution of the trajectories of a continuous time positive system are investigated to this end we need to introduce some new tools and to derive some new results within the broad research area of nonnegative matrix theory which enable use to explore the zero pattern and the elementary modes of the exponential of a metzler matrix specifically a normal form for the exponential of a metzler matrix is provided and the concept of echelon basis consisting of eigenvectors and generalized eigenvectors of the metzler matrix is introduced by making use of these two ingredients a detailed result about the dominant mode of each single block appearing in the normal form of the exponential matrix is provided this allows to obtain a modal decomposition of the exponential matrix emphasizing the column dominant modes as a result the zero pattern as well as the asymptotic behavior of every free state evolution depending on the zero pattern of the initial state can be easily determined

on algebras satisfying x 2x 2 n x x the commutative algebras satisfying the adjoint identity x 2x 2 n x x where n is a cubic form are shown to be related to a class of generically algebraic jordan algebras of degree at most 4 and to the pseudo composition algebras they are classified under a nondegeneracy condition

low rank perturbations and the spectrum of a tridiagonal sign pattern abstract the n by n tridiagonal sign pattern t n has every superdiagonal entry positive every subdiagonal entry negative the 1 1 entry negative the n n entry positive and every other diagonal entry zero inertia and spectral results for matrices a n having the sign pattern t n are proved using new techniques on low rank perturbations it is also shown by using maple that for 8 n 16 t n allows any spectrum these results extend those previously in the literature and strengthen the conjecture that t n allows any spectrum for all values of n in addition bounds on the algebraic multiplicity of an eigenvalue of a low rank perturbation of a general matrix are obtained

peak characteristic and nonnegative signature abstract we introduce the concepts of peak characteristic of an m matrix a and of peak signature and nonnegative signature of subsets of the generalized nullspace e a of a and we discuss the relations between them we also study completions of sets to height bases and to level bases our results are then applied to the study of the nonnegativity of height bases for e a in particular we show that a sequence can serve as the nonnegative signature of some height basis for e a if and only if it is less than or equal to the peak characteristic of a

digraph based determination of jordan block size structure of singular matrix pencils abstract the generic jordan block sizes corresponding to multiple characteristic roots at zero and at infinity of a singular matrix pencil will be determined graph theoretically an application of this technique to detect certain controllability properties of linear time invariant differential algebraic equations is discussed

the contragredient equivalence application to solve some matrix systems in this paper we apply the contragredient equivalence to solve two matrix systems firstly we characterize and build all possible solutions of the matrix system p xy q yx giving a recursive formula for the number of contragrediently nonequivalent solutions and secondly we find the solution of the matrix system ax yc by xd

some applications of spectral theory of nonnegative matrices to input output models abstract we show the necessary and sufficient condition that a nonnegative matrix has a unique positive eigenvector where the analytic expression displaying the linear relations between each remnant component and a basic characteristic subvector of the unique eigenvector is discovered when the nonnegative matrix is reducible as a result we infer the exact necessary and sufficient condition that the iteration matrix m 1 n as a special nonnegative matrix has a unique positive eigenvector when m n is an m splitting which is applied to the condition for the existence and uniqueness of a balanced growth solution for the leontief dynamic input output model previous work in the field did not clearly involve the uniqueness of the balanced growth solution in this paper we develop the prior results that is we find the necessary and sufficient condition that the leontief dynamic input output model has a unique bal anced growth solution finally we obtain the necessary and sufficient condition for theexistence and uniqueness of both the balanced growth solution and the production prices system

a max version of the perron frobenius theorem abstract if a an n n nonnegative irreducible matrix then there exists a 0 and a positive vector x such that maxjaijxj a xi i 1 2 n furthermore a is the maximum geometric mean of a circuit in the weighted directed graph corresponding to a this theorem which we refer to as the max version of the perron frobenius theorem is well known in the context of matrices over the max algebra and also in the context of matrix scalings in the present work which is partly expository we bring out the intimate connection between this result and the perron frobenius theory we present several proofs of the result some of which use the perron frobenius theorem structure of max eigenvalues and max eigenvectors is described possible ways to unify the perron frobenius theorem and its max version are indicated some inequalities for a are proved

a product index theorem with applications to splittings of m matrices abstract let a be an m matrix and b a nonnegative matrix it is shown that with some additional constraints the index of zero for ba is at least as large as the index of zero for a this result is used to develop properties of splittings of an m matrix weak collapsed graph compatible and collapsed graph compatible splittings are introduced here and graph compatible weak graph compatible and block upper triangular splittings are also investigated in the context of z splittings this furthers work done by schneider kavanagh and li

construction and nonnegativity of sign idempotent sign pattern matrices abstract in this paper we modify eschenbach s algorithm for constructing sign idempotent sign pattern matrices so that it correctly constructs all of them we find distinct classes of sign idempotent sign pattern matrices that are signature similar to an entrywise nonnegative sign pattern matrix additionally if for a sign idempotent sign pattern matrix a there exists a signature matrix s such that sas is nonnegative we prove such s is unique up to multiplication by 1 if the signed digraph d a is not disconnected

on a conjecture of fiedler and markham abstract for the hadamard product a a 1 of an m matrix a and its inverse a 1 fiedler and markham conjectured that q a a 1 2 n see m fiedler and t l markham linear algebra appl 101 1988 1 8 where q a a 1 is the smallest eigenvalue in modulus of a a 1 the present paper studies this conjecture an incorrect proof is given in li ching and chen ji cheng linear algebra appl 144 1991 171 178 and establishes q a a 1 2 n n 1 n for some special matrices the conjecture is proved

existence and uniqueness of positive periodic solutions for a neutral logarithmic population model in this paper by using an abstract continuous theorem of k set contractive operator the criteria is established for the existence global attractivity of positive periodic solution of a neutral delay logarithmic population model with multiple delays the result improve the known ones in s p lu w g ge existence of positive periodic solutions for neutral logarithmic population model with multiple delays j comput appl math 166 2 2004 371 383