Supertropical matrix algebra. (English) Zbl 1215.15018

The objective of this paper is to develop a general algebraic theory of a supertropical matrix algebra, extending previous results of the first author. The main results are as follows:
The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible.
There exists an adjoint matrix adj\((A)\) such that the matrix \(A\cdot\text{adj}(A)\) behaves much like the identity matrix (times \(|A|\)).
Every matrix \(A\) is a supertropical root of its Hamilton-Cayley polynomial \(f_A\). If these roots are distinct, then \(A\) is conjugate (in a certain supertropical sense) to a diagonal matrix.
The tropical determinant of a matrix \(A\) is a ghost iff the rows of \(A\) are tropically dependent, iff the columns of \(A\) are tropically dependent.
Every root of \(f_A\) is a “supertropical” eigenvalue of \(A\) (appropriately defined), and has a tangible supertropical eigenvector.


15A30 Algebraic systems of matrices
15A15 Determinants, permanents, traces, other special matrix functions
15A18 Eigenvalues, singular values, and eigenvectors


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[1] M. Akian, R. Bapat and S. Gaubert, Max-plus algebra, in Handbook of Linear Algebra (L. Hogben, R. Brualdi, A. Greenbaum, R. Mathias, eds.), Chapman and Hall, London, 2006. · Zbl 0922.15001
[2] A. Ambrosio, Proof of Birkhoff-von Neumann Theorem, PlanetMath.Org, 2005.
[3] G. Birkhoff, Tres observaciones sobre el algebra lineal, Universidad Nacional de Tucuman Revista, Serie A 5 (1946), 147–151. · Zbl 0060.07906
[4] R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, Cambridge, 1991.
[5] M. Develin, F. Santos and B. Sturmfels, On the tropical rank of a matrix, in Discrete and Computational Geometry, (J. E. Goodman, J. Pach and E. Welzl, eds.), Mathematical Sciences Research Institute Publications, Volume 52, Cambridge University Press, Cambridge, 2005, pp. 213–242. · Zbl 1095.15001
[6] R. Diestel, Graph Theory, Springer, New York, 1997.
[7] A. M. Gibbons, Algorithmic Graph Theory, Cambridge University Press, Cambridge, 1985. · Zbl 0568.05001
[8] J. Golan, The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, Volume 54, Longman Sci & Tech., Harlow, 1992. · Zbl 0780.16036
[9] B. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, New York, 2003. · Zbl 1026.22001
[10] Z. Izhakian, Tropical arithmetic and algebra of tropical matrices, Communications in Algebra 37 (2009), 1445–1468. (Preprint at arXiv:math.AG/0505458). · Zbl 1165.15017
[11] Z. Izhakian, The tropical rank of a matrix, preprint at arXiv:math.AC/0604208, 2005.
[12] Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, preprint, 2009. · Zbl 1225.13009
[13] Z. Izhakian and L. Rowen, Supertropical algebra, Advances in Mathematics 225 (2010), 2222–2286. (Preprint at arXiv:0806.1175.) · Zbl 1273.14132
[14] Z. Izhakian and L. Rowen, The tropical rank of a matrix, Communications in Algebra 37 (2009), 3912–3927. · Zbl 1184.15003
[15] G. Lallement, Semigroups and Combinatorial Applications, Wiley, New York, 1979. · Zbl 0421.20025
[16] J. Richter-Gebert, B. Sturmfels and T. Theobald, First steps in tropical geometry, in Idempotent Mathematics and Mathematical Physics, Proceedings Vienna 2003, (G. L. Litvinov and V. P. Maslov, eds.), Contemporary Mathematics, Vol. 377, American Mathematical Society, Providence, RI, 2005, pp. 289–317. · Zbl 1093.14080
[17] M. Slone, Proof of Hall’s Marriage Theorem, PlanetMath.Org, 2002.
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