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Max-plus definite matrix closures and their eigenspaces. (English) Zbl 1131.15009
This paper provides a contribution to the geometrical understanding of earlier algebraic results on max-plus eigenspaces [cf. P. Butkovič, Discrete Appl. Math. 105, 73–86 (2000; Zbl 0976.15013); P. Butkovič and R. A. Cuninghame-Green, Linear Algebra Appl. 421, No. 2–3, 370–381 (2007; Zbl 1131.15008)] by regarding the max-plus semiring $$\mathbb R _{\max}:= \mathbb R \cup \{-\infty\}$$ with its simplifying total order and using basic tools of max-algebra.
Definite forms for a given matrix $$A= (a_{ij}) \in \mathbb R^{n\times n}_{\max}$$ with nonzero (finite) permanent are considered and it is shown that the closures of all of them coincide, so resulting in the “definite closure” operation. The author presents a description of the eigenspace $$eig(A)$$ and the faces of it for a definite matrix $$A$$. Employing a representation (op. cit.), he concludes that $$eig(A)$$ has a non-empty interior if and only if $$A$$ has a strong permanent. Finally, this interior is fully detailed by means of Hilbert distances.

##### MSC:
 15A18 Eigenvalues, singular values, and eigenvectors 06F15 Ordered groups 15A30 Algebraic systems of matrices 15B33 Matrices over special rings (quaternions, finite fields, etc.)
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##### References:
 [1] Baccelli, F.; Cohen, G.; Olsder, G.-J.; Quadrat, J.-P., Synchronization and linearity. an algebra for discrete event systems, (1992), Wiley New York [2] Butkovič, P., Simple image set of (MAX,+) linear mappings, Discrete appl. math., 105, 73-86, (2000) · Zbl 0976.15013 [3] Butkovič, P., MAX-algebra: linear algebra of combinatorics?, Linear algebra appl., 367, 313-335, (2003) · Zbl 1022.15017 [4] Carré, B.A., An algebra for network routing problems, J. inst. math. appl., 7, 273-299, (1971) · Zbl 0219.90020 [5] Cohen, G.; Gaubert, S.; Quadrat, J.-P., Duality and separation theorems in idempotent semimodules, Linear algebra appl., 379, 395-422, (2004), Available from: · Zbl 1042.46004 [6] Cuninghame-Green, R.A., Minimax algebra, Lecture notes in economics and mathematical systems, vol. 166, (1979), Springer Berlin [7] Cuninghame-Green, R.A.; Butkovič, P., Bases in MAX-algebra, Linear algebra appl., 389, 107-120, (2004) · Zbl 1059.15001 [8] Develin, M.; Santos, F.; Sturmfels, B., On the rank of a tropical matrix, (), 213-242, Available from: · Zbl 1095.15001 [9] Develin, M.; Sturmfels, B., Tropical convexity, Doc. math., 9, 1-27, (2004), Available from: · Zbl 1054.52004 [10] S. Gaubert, Théorie des Systèmes Linéaires dans les Dioïdes, Thèse, Ecole des Mines des Paris, Paris, 1992. [11] Kolokoltsov, V.N.; Maslov, V.P., Idempotent analysis and its applications, (1997), Kluwer Academic Publishers Dordrecht · Zbl 0941.93001 [12] G.L. Litvinov, V.P. Maslov, Correspondence principle for idempotent calculus and some computer applications, Bures-Sur-Yvette: Institut des Hautes Etudes Scientifiques (IHES/M/95/33), 1995; See also: J. Gunawardena (Ed.), Idempotency, Publ. of the I. Newton Institute, Cambridge University Press, 1998, pp. 420-443. · Zbl 0897.68050 [13] Litvinov, G.L.; Maslov, V.P.; Shpiz, G.B., Idempotent functional analysis: an algebraical approach, Math. notes, 69, 5, 696-729, (2001), Available from: · Zbl 1017.46034 [14] Litvinov, G.L.; Maslova, E.V., Universal numerical algorithms and their software implementation, Program. comput. software, 26, 5, 275-280, (2000) · Zbl 0968.68194 [15] P. Moller, Théorie Algébraique des Systèmes à Evénements Discrets, Thèse, Ecole des Mines des Paris, Paris, 1988. [16] Rote, G., A systolic array algorithm for the algebraic path problem, Computing, 34, 191-219, (1985) · Zbl 0562.68056 [17] E. Wagneur, Moduloids and Pseudomodules-1-dimension theory, in: J.L. Lions, A. Bensoussan (Eds.), Analysis and Optimization of Systems, Lecture Notes in Control and Information Sciences, 1988. · Zbl 0757.06008 [18] Wagneur, E., Moduloids and pseudomodules-1-dimension theory, Discrete math., 98, 57-73, (1991) · Zbl 0757.06008 [19] Zimmermann, U., Linear and combinatorial optimization in ordered algebraic structures, (1981), North Holland Amsterdam · Zbl 0466.90045
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