×

Dependence of supertropical eigenspaces. (English) Zbl 1378.15006

This paper is devoted to the analysis of the issues that causes tropical eigenspaces of distinct supertropical eigenvalues of a nonsingular matrix A to be dependent. Starting with a long survey of the recent results of tropical polynomials, the authors introduce properties of matrices and vectors in the tropical structure, with definitions extended to the supertropical framework. Next, they analyze the dependence between eigenvectors, using their definition according to the so-called eigenvectors algorithm described by Z. Izhakian and L. Rowen [Isr. J. Math. 186, 69–96 (2011; Zbl 1277.15013)]. Some special cases are resolved. In the end, sufficient conditions for independence are stated, and two conjectures on the eigenvectors of the quasi-inverse of a matrix are proposed.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A09 Theory of matrix inversion and generalized inverses
15A15 Determinants, permanents, traces, other special matrix functions
15A80 Max-plus and related algebras
15B33 Matrices over special rings (quaternions, finite fields, etc.)

Citations:

Zbl 1277.15013
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Akian, M., Bapat, R., Gaubert, S. (2006). Max-plus algebra. In: Hogben, L., Brualdi, R., Greenbaum, A., Mathias, R., eds.Handbook of Linear Algebra. London: Chapman and Hall, pp. 25.1–25.17.
[2] DOI: 10.1090/conm/495/09689
[3] Akian M., Tropical polyhedra are equivalent to mean payoff games. J. Algebra Comput. (2012) · Zbl 1239.14054
[4] Akian, M., Gaubert, S., Walsh, C. (2005). Discrete max-plus spectral theory. In: Litvinov, G. L., Maslov, V. P., eds.Idem. Math. and Mathematical Physics. Contemporary Math., Vol. 377. AMS, pp. 53–77. · Zbl 1104.47055
[5] DOI: 10.1016/j.laa.2013.12.021 · Zbl 1297.14063
[6] Akian, M., Gaubert, S., Niv, A. (2015). Tropical compound matrix identities. Preprint.
[7] Butkovic P., Kybernetika 39 (2) pp 129– (2001)
[8] DOI: 10.1016/S0024-3795(02)00655-9 · Zbl 1022.15017
[9] Butkovic P., CEJOR 8 pp 237– (2000)
[10] Fiedler M., Linear Optimization Problems with Inexact Data (2006) · Zbl 1106.90051
[11] Gaubert S., Théorie des systèmes linéaires dans les dioïdes (1992)
[12] Gaubert, S., Sharify, M. (2009).Tropical Scaling of Polynomial Matrices. Lecture Notes in Control and Information Sciences, Vol. 389. Springer, pp. 291–303. · Zbl 1186.15007
[13] Gondran, M. (1975). Path algebra and algorithms. In: Roy, B. ed.Combinatorial Programing: Methods and Applications. Dordrecht: Reidel, pp. 137–148. · Zbl 0326.90060
[14] Itenberg I., Tropical Algebraic Geometry (2007) · Zbl 1162.14300
[15] DOI: 10.1080/00927870802466967 · Zbl 1165.15017
[16] DOI: 10.1016/j.jpaa.2011.01.002 · Zbl 1225.13009
[17] Izhakian Z., J. An. Ştiinţ. Univ. ”Ovidius” Constanţa, Ser. Mat. 19 (2) pp 131– (2011)
[18] DOI: 10.2140/pjm.2013.266.43 · Zbl 1396.15021
[19] DOI: 10.1016/j.aim.2010.04.007 · Zbl 1273.14132
[20] DOI: 10.1007/s11856-011-0036-2 · Zbl 1215.15018
[21] DOI: 10.1007/s11856-011-0133-2 · Zbl 1277.15013
[22] DOI: 10.1016/j.jalgebra.2011.06.002 · Zbl 1283.15055
[23] Jonczy J., Algebraic Path Problems (2008)
[24] DOI: 10.1070/RM1996v051n06ABEH003011 · Zbl 0916.49019
[25] DOI: 10.1090/conm/495 · Zbl 1172.00019
[26] Mikhalkin, G. (2006). Tropical geometry and its applications. In:Proceedings of the ICM, Vol. II. Zürich: European Mathematical Society, pp. 827–852. (arXiv: [math. AG]0601041v2). · Zbl 1103.14034
[27] DOI: 10.1080/00927872.2012.717324 · Zbl 1302.15008
[28] Niv A., On Pseudo-inverses of matrices and their characteristic polynomials in supertropical algebra 471 pp 264– (2015) · Zbl 1310.15009
[29] DOI: 10.1016/j.laa.2006.02.038 · Zbl 1131.15009
[30] Straubing, H. (1983). A combinatorial proof of the Cayley-Hamilton theorem.Discrete Math.43(2–3):273–279. · Zbl 0533.15010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.