Linear Projectors in the max-plus Algebra G. Cohen S. Gaubert J.-P. Quadrat Centre Automatique et Systemes` INRIA Ecole´ des Mines de Paris Rocquencourt Fontainebleau, France Le Chesnay, France
[email protected] Stephane.Gaubert Jean-Pierre.Quadrat @inria.fr Abstract In this paper, we consider the linear projection problem. Consider three semimodules , , and two linear maps In general semimodules, we say that the image of a linear B, C: U X Y operator B and the kernel of a linear operator C are direct B C factors if every equivalence class modulo C crosses the . (2) image of B at a unique point. For linear maps represented U → X → Y by matrices over certain idempotent semifields such as the We say that im B and ker C are direct factors if for all x (max, )-semiring, we give necessary and sufficient con- , there is a unique im B such that Cx C. When∈ ditions+ for an image and a kernel to be direct factors. We X ∈ C = it is the case: 1. the map 5B : , x z, which is characterize the semimodules that admit a direct factor (or C 2 C XC → X 7→C C linear, satisfies (5B ) 5B , 5B B B, C5B C (5B equivalently, the semimodules that are the image of a lin- is the projector onto im=B, parallel to= ker C); 2.= we have ear projector): their matrices have a g-inverse. We give the isomorphism / ker C im B; in particular, if and simple effective tests for all these properties, in terms of are free finitelyX generated' semimodules, then theU linear matrix residuation.