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

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

optimal control of dynamic systems application to spline approximations generally classical polynomial splines tend to exhibit unwanted undulations in this work we discuss a technique based on control principles for eliminating these undulations and increasing the smoothness properties of the spline interpolants we give a generalization of the classical polynomial splines and show that this generalization is in fact a family of splines that covers the broad spectrum of polynomial trigonometric and exponential splines a particular element in this family is determined by the appropriate control data it is shown that this technique is easy to implement several numerical and curve fitting examples are given to illustrate the advantages of this technique over the classical approach finally we discuss the convergence properties of the interpolant

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

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

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

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

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

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

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

the significance of the evanescent spectrum in structure waveguide interaction problems modal decomposition is often applied in elastodynamics and acoustics for the solution of problems related to propagation of mechanical disturbances in waveguides one of the key elements of this method is the solution of an eigenvalue problem for obtaining the roots of the dispersion equation which signify the wavenumbers of the waves that may exist in the system for non dissipative media the wavenumber spectrum consists of a finite number of real roots supplemented by infinitely many imaginary and complex ones the former refer to the propagating modes in the medium whereas the latter constitute the so called evanescent spectrum this study investigates the significance of the evanescent spectrum in structure waveguide interaction problems two cases are analysed namely a beam in contact with a fluid layer and a cylindrical shell interacting with an acousto elastic waveguide the first case allows the introduction of a modal decomposition method and the establishment of appropriate criteria for the truncation of the modal expansions in a simple mathematical framework the second case describes structure borne wave radiation in an offshore environment during the installation of a pile with an impact hammer a problem that has raised serious concerns in recent years due to the associated underwater noise pollution

enclosing all zeros of a system of analytic functions abstract we present a rigorous numerical method for location of simple zeros of a system of two analytic functions in a rectangular cuboid domain based on the logarithmic integral we compare this to a simpler also rigorous method based on bisection the latter is determined to be more efficient in the examples considered this is mainly due to inefficient methods for computing the logarithmic integral occurring in the former method

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

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 parameter power series for sturm liouville equations with a potential polynomially dependent on the spectral parameter and zakharov shabat systems a spectral parameter power series spps representation for solutions of sturm liouville equations of the form pu qu u k 1n krk is obtained such equations are known as sturm liouville equations with potentials polynomially dependent on the spectral parameter or sturm liouville pencils slp it allows one to write a general solution of the equation as a power series in terms of the spectral parameter the coefficients of the series are given in terms of recursive integrals involving a particular solution of the equation pu0 qu0 0 the convenient form of the solution of slp provides an efficient numerical method for solving corresponding initial value boundary value and spectral problems a special case of the slp arises in relation with the zakharov shabat system we derive a spps representation for its general solution and consider other applications as the one dimensional dirac system and the equation describing a damped string several numerical examples illustrate the efficiency and the accur

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

liouville transformation analytic approximation of transmutation operators and solution of spectral problems a method for solving spectral problems for the sturm liouville equation p v q v r v 0 based on the approximation of the delsarte transmutation operators combined with the liouville transformation is presented the problem of numerical approximation of solutions and of eigendata is reduced to approximation of a pair of functions depending on the coefficients p q and r by a finite linear combination of certain specially constructed functions related to generalized wave polynomials introduced by khmelnytskaya et al 2013 and kravchenko and torba 2015 the method allows one to compute both lower and higher eigendata with an extreme accuracy several necessary results concerning the action of the liouville transformation on formal powers arising in the method of spectral parameter power series are obtained as well as the transmutation operator for the sturm liouville operator 1 r d d x p d d x q

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

a reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain the argument principle is a natural and simple method to determine the number of zeros of an analytic functionf z in a bounded domaind n the number of zeros counting multiplicities off z is 1 2 times the change in argf z asz moves along the closed contour d since the range of argf z is a critical point in the computational procedure is to assure that the discretization of d z i i 1 m is such that delta text z i text text z i 1 text arg f z leqq pi discretization control which may violate this inequality can lead to an unreliable algorithm mathematical theorems derived for the discretization of d lead to a completely reliable algorithm to computen this algorithm also treats in an elementary way the case when a zero is on or near the contour d numerical examples are given for the reliable algorithm formulated here and it is pointed out in these examples how inadequate discretization control can lead to failure of other algorithms

the use of spline on spline for the approximation of cauchy principal value integrals in this paper the spline on spline interpolatory polynomials are used to approximate cauchy principal value integrals and their derivatives this technique yields better results than using the derivatives of the traditional single spline functions which has been considered by orsi improved order of convergence for these approximations are discussed

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

enclosing all zeros of an analytic function a rigorous approach we present a method to find all zeros of an analytic function in a rectangular domain the approach is based on finding guaranteed enclosures rather than approximations of the zeros well isolated simple zeros are determined fast and with high accuracy clusters of zeros can in many cases be distinguished from multiple zeros by applying the argument principle to sufficiently high order derivatives of the function we illustrate the proposed method through five examples of varying levels of complexity

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

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

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

root finding in the complex plane for seismo acoustic propagation scenarios with green s function solutions a normal mode solution to the ocean acoustic problem of the pekeris waveguide with an elastic bottom using a green s function formulation for a compressional wave point source is considered analytic solutions to these types of waveguide propagation problems are strongly dependent on the eigenvalues of the problem these eigenvalues represent horizontal wavenumbers corresponding to propagating modes of energy the eigenvalues arise as singularities in the inverse hankel transform integral and are specified by roots to a characteristic equation these roots manifest themselves as poles in the inverse transform integral and can be both subtle and difficult to determine following methods previously developed s ivansson et al j sound vib 161 1993 a root finding routine has been implemented using the argument principle using the roots to the characteristic equation in the green s function formulation full field solutions are calculated for scenarios where an acoustic source lies in either the water column or elastic half space solutions are benchmarked against laboratory data and existing numerical solutions

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

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

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

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

locating all the zeros of an analytic function in one complex variable based on the argument principle we propose an adaptive multilevel subdivision algorithm for the computation of all the zeros of an analytic function f c c within a bounded domain we illustrate the reliability of this method by several numerical examples

analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems a method for approximate solution of spectral problems for sturm liouville equations based on the construction of the delsarte transmutation operators is presented in fact the problem of numerical approximation of solutions and eigenvalues is reduced to approximation of a primitive of the potential by a finite linear combination of generalized wave polynomials introduced in khmelnytskaya et al 2013 and kravchenko and torba 2014 the method allows one to compute both lower and higher eigendata with an extreme accuracy

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

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

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

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

mapping based algorithm for large scale computation of quasi polynomial zeros a method for computing all zeros of a retarded quasi polynomial that are located in a large region of the complex plane is presented the method is based on mapping the quasi polynomial and on utilizing asymptotic properties of the chains of zeros first the asymptotic exponentials of the chains are determined based on the distribution diagram of the quasi polynomial secondly large regions free of zeros are defined finally the zeros are located as the intersection points of the zero level curves of the real and imaginary parts of the quasi polynomial which are evaluated over the areas of the region outside those free of zeros

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

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

finite difference end conditions and semi cardinal interpolation with quintic b splines abstract for quintic spline interpolation on a uniform grid we formulate a class of end conditions in terms of homogeneous finite differences of b spline coefficients the analysis of the resulting schemes including their approximation order is based on the model of semi cardinal interpolation at the set of non negative integers numerical results are also presented to illustrate the implementation on a finite grid

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

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

overconvergence properties of quintic interpolatory splines let q be a quintic spline with equi spaced knots on a b interpolating a given function y at the knots the parameters which determine q are used to construct a piecewise defined polynomial p of degree six it is shown that p can be used to give at any point of a b better orders of approximation to y and its derivatives than those obtained from q it is also shown that the superconvergence properties of the derivatives of q at specific points of a b are all simple consequences of the properties of p