The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Lecture Notes in Economics and Mathematical Systems

Max-algebra: Two Special Features.- One-sided Max-linear Systems and Max-algebraic Subspaces.- Eigenvalues and Eigenvectors.- Maxpolynomials. The Characteristic Maxpolynomial.- Linear Independence and Rank. The Simple Image Set.- Two-sided Max-linear Systems.- Reachability of Eigenspaces.- Generalized Eigenproblem.- Max-linear Programs.- Conclusions and Open Problems.

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Max-linear Systems: Theory and Algorithms

Four relations are defined the use of which allow a reliability network to be expressed as a mathematical statement. Series and active and standby redundancy connections are allowed. The possibility of requiring more than one of the active branches to be operative is included. At the very least the result allows for typing in place of drawing a network description.

http://ieeexplore.ieee.org/iel5/24/5216426/05216446.pdf

Network Calculus

We investigate different notions of linear independence and of ma- trix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a no- tion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples.

Linear independence over tropical semirings and beyond

https://hal.inria.fr/inria-00071358/document

The Minkowski theorem for max-plus convex sets

The concept of (A,B)-invariant subspace (or controlled invariant) of a linear dynamical system is extended to linear systems over the max-plus semiring Although this extension presents several difficulties, which are similar to those encountered in the same kind of extension to linear dynamical systems over rings, it appears capable of providing solutions to many control problems like in the cases of linear systems over fields or rings Sufficient conditions are given for computing the maximal (A,B)-invariant subspace contained in a given space and the existence of linear state feedbacks is discussed An application to the study of transportation networks which evolve according to a timetable is considered

https://hal.inria.fr/inria-00070485/document

Max-Plus $(A,B)$ -Invariant Spaces and Control of Timed Discrete-Event Systems

We extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max-plus or tropical semiring. We establish a duality theorem relating both notions, which we use to construct dynamic observers. These are useful in situations in which some of the system coefficients may vary within certain intervals. The results are illustrated by an application to a manufacturing system.

Duality Between Invariant Spaces for Max-Plus Linear Discrete Event Systems

http://web.mat.bham.ac.uk/P.Butkovic/My%20grant%20papers/LFP%20paper%20by%20sergei%20et%20al.pdf

Tropical linear-fractional programming and parametric mean payoff games

We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representation of a polyhedron turns out to be provided by the extreme elements of its tropical polar. We characterize these extreme elements, showing in particular that they are determined by support vectors.

/pdf/minimal-half-spaces-and-external-representation-of-tropical-1q5vfoiird.pdf

Ricardo D. Katz

Papers

The Minkowski theorem for max-plus convex sets

Max-Plus $(A,B)$ -Invariant Spaces and Control of Timed Discrete-Event Systems

Duality Between Invariant Spaces for Max-Plus Linear Discrete Event Systems

Tropical linear-fractional programming and parametric mean payoff games

Minimal half-spaces and external representation of tropical polyhedra