on pseudo inverses of matrices and their characteristic polynomials in supertropical algebra abstract the only invertible matrices in tropical algebra are diagonal matrices permutation matrices and their products however the pseudo inverse a defined as 1 det a adj a with det a being the tropical permanent also called the tropical determinant of a matrix a inherits some classical algebraic properties and has some surprising new ones defining b and b to be tropically similar if b a b a we examine the characteristic max polynomials of tropically similar matrices as well as those of pseudo inverses other miscellaneous results include a new proof of the identity for det a b and a connection to stabilization of the powers of definite matrices

x simple image eigencones of tropical matrices abstract we investigate max algebraic tropical one sided systems a x b where b is an eigenvector and x lies in an interval x a matrix a is said to have x simple image eigencone associated with an eigenvalue if any eigenvector x associated with and belonging to the interval x is the unique solution of the system a y x in x we characterize matrices with x simple image eigencone geometrically and combinatorially and for some special cases derive criteria that can be efficiently checked in practice

max algebra the linear algebra of combinatorics abstract let a b max a b a b a b for a b r r by max algebra we understand the analogue of linear algebra developed for the pair of operations extended to matrices and vectors max algebra which has been studied for more than 40 years is an attractive way of describing a class of nonlinear problems appearing for instance in machine scheduling information technology and discrete event dynamic systems this paper focuses on presenting a number of links between basic max algebraic problems like systems of linear equations eigenvalue eigenvector problem linear independence regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimisation problems on the other hand this indicates that max algebra may be regarded as a linear algebraic encoding of a class of combinatorial problems the paper is intended for wider readership including researchers not familiar with max algebra

fiedler ptak scaling in max algebra abstract this is essentially the text of my talk on fiedler ptak scaling in max algebra delivered in the invited minisymposium in honor of miroslav fiedler at the 17th ilas conference in braunschweig germany

identification of continuous time arx models from irregularly sampled data the problem of estimating the parameters in a continuous time arx process from unevenly sampled data is studied a solution where the differentiation operator is replaced by a difference operator is suggested in the paper results are given for how the difference operator should be chosen in order to obtain consistent parameter estimates the proposed method is considerably faster than conventional methods such as the maximum likelihood method the crameacuter rao bound for estimation of the parameters is computed in the derivation the slepian bangs formula is used together with a state space framework resulting in a closed form expression for the crameacuter rao bound numerical studies indicate that the crameacuter rao bound is reached by the proposed method

max algebraic attraction cones of nonnegative irreducible matrices abstract it is known that the max algebraic powers ar of a nonnegative irreducible matrix are ultimately periodic this leads to the concept of attraction cone attr a t by which we mean the solution set of a two sided system t a ar x ar t x where r is any integer after the periodicity transient t a and a is the maximum cycle geometric mean of a a question which this paper answers is how to describe attr a t by a concise system of equations without knowing t a this study requires knowledge of certain structures and symmetries of periodic max algebraic powers which are also described we also consider extremals of attraction cones in a special case and address the complexity of computing the coefficients of the system which describes attraction cone

on sets of eigenvalues of matrices with prescribed row sums and prescribed graph abstract motivated by a work of boros brualdi crama and hoffman we consider the sets of i possible perron roots of nonnegative matrices with prescribed row sums and associated graph and ii possible eigenvalues of complex matrices with prescribed associated graph and row sums of the moduli of their entries to characterize the set of perron roots or possible eigenvalues of matrices in these classes we introduce following an idea of al pin elsner and van den driessche the concept of row uniform matrix which is a nonnegative matrix where all nonzero entries in every row are equal furthermore we completely characterize the sets of possible perron roots of the class of nonnegative matrices and the set of possible eigenvalues of the class of complex matrices under study extending known results to the reducible case we derive new sharp bounds on the set of eigenvalues or perron roots of matrices when the only information available is the graph of the matrix and the row sums of the moduli of its entries in the last section of the paper a new constructive proof of the camion hoffman theorem is given

reachability of eigenspaces for interval circulant matrices in max algebra abstract a nonnegative matrix a is said to be strongly robust if its max algebraic eigencone is universally reachable i e if the orbit of any initial vector ends up with a max algebraic eigenvector of a consider the case when the initial vector is restricted to an interval and a can be any matrix from a given interval of nonnegative circulant matrices the main aim of this paper is to classify and characterize the six types of interval robustness in this situation this naturally leads us also to study the max algebraic spectral theory of circulant matrices and the relation of inclusion between attraction cones of circulant matrices in max algebra

matrices commuting with a given normal tropical matrix abstract consider the space m n nor of square normal matrices x x i j over r i e x i j 0 and x i i 0 endow m n nor with the tropical sum and multiplication fix a real matrix a m n nor and consider the set a of matrices in m n nor which commute with a we prove that a is a finite union of alcoved polytopes in particular a is a finite union of convex sets the set a a of x such that a x x a a is also a finite union of alcoved polytopes the same is true for the set a of x such that a x x a x a topology is given to m n nor then the set a a is a neighborhood of the identity matrix i if a is strictly normal then a is a neighborhood of the zero matrix in one case a is a neighborhood of a we give an upper bound for the dimension of a we explore the relationship between the polyhedral complexes span a span x and span a x when a and x commute two matrices denoted a and a arise from a in connection with a the geometric meaning of them is given in detail for one example we produce examples of matrices which commute in any dimension

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

tropical totally positive matrices abstract we investigate the tropical analogues of totally positive and totally nonnegative matrices these arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field like the field of real puiseux series we show that the nonarchimedean valuation sends the totally positive matrices precisely to the monge matrices this leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices we also show that tropical totally nonnegative matrices with a finite permanent can be factorized in terms of elementary matrices we finally determine the eigenvalues of tropical totally nonnegative matrices and relate them with the eigenvalues of totally nonnegative matrices over nonarchimedean fields

modifying the power method in max algebra in the max algebra system the eigenequation for an n n irreducible nonnegative matrix a aij is a x a x here a x i maxjaijxj and a is the maximum circuit geometric mean the complexity of the power method given in l elsner p van den driessche linear algebra appl to appear to compute a and x is considered under some assumptions on the critical matrix it is shown that the algorithm may have time complexity o n4 a modified power method based on karp s formula is presented for this new algorithm with no assumptions on the critical matrix a and x can be computed in o n3 time furthermore this algorithm can be used to compute all linearly independent eigenvectors corresponding to a

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

a polynomial algorithm for solving system of inequalities in max plus algebra abstract this paper deals with the system of inequalities a x x in which a is a square matrix x is a column vector and is a max plus composition it provides two algorithms which can be used to find a finite proper supereigenvectors of a corresponding to the eigenvalue 0 in a case of irreducible matrix a with the maximum cycle mean a 0 and a finite solution of the system in a case of reducible matrix a respectively in a polynomial number of steps

the analytic hierarchy process max algebra and multi objective optimisation the analytic hierarchy process ahp is widely used for decision making involving multiple criteria elsner and van den driessche 2004 2010 10 11 introduced a max algebraic approach to the single criterion ahp we extend this to the multi criteria ahp by considering multi objective generalisations of the single objective optimisation problem solved in these earlier papers we relate the existence of globally optimal solutions to the commutativity properties of the associated matrices we relate min max optimal solutions to the generalised spectral radius and we prove that pareto optimal solutions are guaranteed to exist

the ultimate rank of tropical matrices a tropical matrix is a matrix defined over the max plus semiring for such matrices there exist several non coinciding notions of rank the row rank the column rank the schein barvinok rank the kapranov rank or the tropical rank among others in the present paper we show that there exists a natural notion of ultimate rank for the powers of a tropical matrix which does not depend on the underlying notion of rank furthermore we provide a simple formula for the ultimate rank of a matrix which can therefore be computed in polynomial time then we turn our attention to finitely generated semigroups of matrices for which our notion of ultimate rank is generalized naturally we provide both combinatorial and geometric characterizations of semigroups having maximal ultimate rank as a consequence we obtain a polynomial algorithm to decide if the ultimate rank of a finitely generated semigroup is maximal

optimal assignments with supervisions abstract in this paper we provide a new graph theoretic proof of the tropical jacobi identity recently obtained in 1 we also develop an application of this theorem to optimal assignments with supervisions that is optimally assigning multiple tasks to one team or daily tasks to multiple teams where each team has a supervisor task or a supervised task

weak csr expansions and transience bounds in max plus algebra abstract this paper aims to unify and extend existing techniques for deriving upper bounds on the transient of max plus matrix powers to this aim we introduce the concept of weak csr expansions a t c s t r b t we observe that most of the known bounds implicitly take the maximum of i a bound for the weak csr expansion to hold which does not depend on the values of the entries of the matrix but only on its pattern and ii a bound for the c s t r term to dominate to improve and analyze i we consider various cycle replacement techniques and show that some of the known bounds for indices and exponents of digraphs apply here we also show how to make use of various parameters of digraphs to improve and analyze ii we introduce three different kinds of weak csr expansions named after nachtigall hartman arguelles and cycle threshold as a result we obtain a collection of bounds in general incomparable to one another but better than the bounds found in the literature

on tropical supereigenvectors abstract the task of finding tropical eigenvectors and subeigenvectors that is non trivial solutions to a x x and a x x in the max plus algebra has been studied by many authors since the 1960s in contrast the task of finding supereigenvectors that is solutions to a x x has attracted attention only recently we present a number of properties of supereigenvectors focusing on a complete characterization of the values of associated with supereigenvectors and in particular finite supereigenvectors the proof of the main statement is constructive and enables us to find a non trivial subspace of finite supereigenvectors we also present an overview of key related results on eigenvectors and subeigenvectors

resource optimization and min spectral theory shows that certain resource optimization problems relative to timed event graphs reduce to linear programs the auxiliary variables which allow this reduction can be interpreted in terms of eigenvectors in the min algebra

an algorithm to describe the solution set of any tropical linear system a x b x abstract an algorithm to give an explicit description of all the solutions to any tropical linear system a x b x is presented the given system is converted into a finite rather small number p of pairs s t of classical linear systems a system s of equations and a system t of inequalities the notion introduced here that makes p small is called compatibility the particular feature of both s and t is that each item equation or inequality is bivariate i e it involves exactly two variables one variable with coefficient 1 and the other one with 1 s is solved by gaussian elimination we explain how to solve t by a method similar to gaussian elimination to achieve this we introduce the notion of sub special matrix the procedure applied to t is therefore called sub specialization

rank functions of tropical matrices abstract we consider rank functions important in tropical linear algebra the tropical rank equal to the topological dimension of the tropical linear span of the columns of a given matrix the factor rank equal to the smallest number of vectors containing these columns in their span the kapranov rank related to problems of tropical algebraic geometry and the determinantal and gondran minoux ranks which generalize the classical linear algebraic notion of rank we are interested in studying the arithmetic properties of the rank functions their mutual behavior and the related computational problems we discuss different open problems related to rank functions of tropical matrices and illustrate them by the number of examples

subeigenvectors and supereigenvectors of fuzzy matrices abstract a fuzzy algebra is a triple b where b is a nonempty bounded linearly ordered set and a b max a b a b min a b for a b b a vector x is said to be a eigenvector of a square matrix a if a x x for some b the aim of the paper is to solve subeigenproblem and supereigenproblem for some b that is to find a solution x of a x x and a x x x is called subeigenvector and supereigenvector respectively the problems are related to and motivated by the similar problems of tropical linear algebra in this paper the properties of subeigenvectors and supereigenvectors are described and the values associated with subeigenvectors and supereigenvectors are characterized as a consequence efficient algorithms for checking all equivalent conditions are introduced

duality and interval analysis over idempotent semirings abstract in this paper semirings with an idempotent addition are considered these algebraic structures are endowed with a partial order this allows to consider residuated maps to solve systems of inequalities a x b see 3 the purpose of this paper is to consider a dual product denoted and the dual residuation of matrices in order to solve the following inequality a x x b x sufficient conditions ensuring the existence of a non linear projector in the solution set are proposed the results are extended to semirings of intervals such as they were introduced in 25

positive path product and inverse m matrices it is known that an inverse m matrix is strict path product but not necessarily vice versa for n 3 it is shown that any square positive matrix may be made strict path product by predictable additions to the diagonal and that any normalized strict path product matrix may be made inverse m by additions to the diagonal that are bounded in terms of the size of the matrix the latter has implications for pairs of inverse m matrices whose hadamard product is inverse m a determinantal inequality relating principal minors and certain associated almost principal minors is derived for normalized inverse m matrices

the tropical commuting variety we study tropical commuting matrices from two viewpoints linear algebra and algebraic geometry in classical linear algebra there exist various criteria to test whether two square matrices commut

a congruence on the semiring of normal tropical matrices abstract we introduce and study a congruence on the normal tropical matrix semiring m n n which is relevant to the kleene stars of normal tropical matrices we prove that this congruence is a bisemilattice congruence and give an exact description of each class in particular we show that the class e is an interval when e is a strongly regular normal tropical matrix we also present a method using floyd warshall algorithm to compute the greatest lower bound of e in such case

max balancing weighted directed graphs and matrix scaling a weighted directed graph g is a triple v a g where v a is a directed graph and g is an arbitrary real valued function defined on the arc set a let g be a strongly connected simple weighted directed graph we say that g is max balanced if for every nontrivial subset of the vertices w the maximum weight over arcs leaving w equals the maximum weight over arcs entering w we show that there exists a up to an additive constant unique potential pv for v v such that v a gp is max balanced where gap pu ga pv for a u v a we describe an o v 2 a algorithm for computing p using an algorithm for computing the maximum cycle mean of g finally we apply our principal result to the similarity scaling of nonnegative matrices

on visualization scaling subeigenvectors and kleene stars in max algebra abstract the purpose of this paper is to investigate the interplay arising between max algebra convexity and scaling problems the latter which have been studied in nonnegative matrix theory are strongly related to max algebra one problem is that of strict visualization scaling defined as for a given nonnegative matrix a a diagonal matrix x such that all elements of x 1 ax are less than or equal to the maximum cycle geometric mean of a with strict inequality for the entries which do not lie on critical cycles in this paper such scalings are described by means of the max algebraic subeigenvectors and kleene stars of nonnegative matrices as well as by some concepts of convex geometry

the minkowski theorem for max plus convex sets abstract we establish the following max plus analogue of minkowski s theorem any point of a compact max plus convex subset of r n can be written as the max plus convex combination of at most n 1 of the extreme points of this subset we establish related results for closed max plus cones and closed unbounded max plus convex sets in particular we show that a closed max plus convex set can be decomposed as a max plus sum of its recession cone and of the max plus convex hull of its extreme points

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

max algebraic powers of irreducible matrices in the periodic regime an application of cyclic classes abstract in max algebra it is well known that the sequence of max algebraic powers a k with a an irreducible square matrix becomes periodic after a finite transient time t a and the ultimate period is equal to the cyclicity of the critical graph of a in this connection we study computational complexity of the following problems 1 for a given k compute a periodic power a r with r k mod and r t a 2 for a given x find the ultimate period of a l x we show that both problems can be solved by matrix squaring in o n 3 log n operations the main idea is to apply an appropriate diagonal similarity scaling a x 1 ax called visualization scaling and to study the role of cyclic classes of the critical graph

on the integration schemes for drucker prager s elastoplastic models based on exponential maps rate plasticity equations for the case of drucker prager s model in small strain regime are considered by defining an augmented stress vector the formulations convert the original problem into a quasi linear differential equation system two new exponential mapping schemes for integrating model equations are proposed in addition two traditional schemes for solving the dynamical system in an explicit manner are discussed the two semi implicit schemes developed pose higher accuracy and better convergency error contours are provided for all four methods to display the accuracy of each scheme in order to compare the results these contours for the classical one step backward euler integration method are also displayed accuracy and efficiency along with the rate of convergency of the existing and the proposed methods are examined by numerical examples copyright 2007 john wiley sons ltd

on the power method in max algebra abstract the eigenvalue problem for an irreducible nonnegative matrix a a ij in the max algebra system is a otimes x lambda x where a otimes x i mathop max j a ij x j and lambda turns out to be the maximum circuit geometric mean mu a a power method algorithm is given to compute mu a and eigenvector x the algorithm is developed by using results on the convergence of max powers of a which are proved using nonnegative matrix theory in contrast to an algorithm developed in 4 this new method works for any irreducible nonnegative a and calculates eigenvectors in a simpler and more efficient way some asymptotic formulas relating mu a the spectral radius and norms are also given

recognizing weakly stable matrices a max plus matrix a is called weakly stable if the sequence orbit x a otimes x a 2 otimes x ldots does not reach an eigenvector of a for any x unless x is an eigenvector this is in contrast to previously studied strongly stable robust matrices for which the orbit reaches an eigenvector with any nontrivial starting vector max plus matrices are used to describe multiprocessor interactive systems for which reachability of a steady regime is equivalent to reachability of an eigenvector by a matrix orbit we prove that an irreducible matrix is weakly stable if and only if its critical graph is a hamiltonian cycle in the associated graph we extend this condition to reducible matrices these criteria can be checked in polynomial time

convexity of tropical polytopes abstract we study the relationship between min plus max plus and euclidean convexity for subsets of r n we introduce a construction which associates to any max plus convex set with compact projectivisation a canonical matrix called its dominator the dominator is a kleene star whose max plus column space is the min plus convex hull of the original set we apply this to show that a set which is any two of i a max plus polytope ii a min plus polytope and iii a euclidean polytope must also be the third in particular these results answer a question of sergeev schneider and butkovic 16 and show that row spaces of tropical kleene star matrices are exactly the polytropes studied by joswig and kulas 13

max plus singular values in this paper we prove a new characterization of the max plus singular values of a max plus matrix as the max plus eigenvalues of an associated max plus matrix pencil this new characterization allows us to compute max plus singu lar values quickly and accurately as well as capturing the asymptotic behavior of the singular values of classical matrices whose entries are exponentially pa rameterized we show experimentally that max plus singular values give order of magnitude approximations to the classical singular values of parameter inde pendent classical matrices we also discuss hungarian scaling which is a diagonal scaling strategy for preprocessing classical linear systems we show that hungarian scaling can dramatically reduce the 2 norm condition number and that this action can be explained using our new theory for max plus singular values keywords tropical algebra max plus algebra singular values diagonal scaling condition number optimal assignment problem we also discuss hungarian scaling which is a diagonal scaling strategy for preprocessing classical linear systems we show that hungarian scaling can dramatically reduce the d norm condition number and that this action can be explained using our new theory for max plus singular values