Abstract
We continue the study of matrices over a supertropical algebra, proving the existence of a tangible adjoint of A, which provides the unique right (resp. left) quasi-inverse maximal with respect to the right (resp. left) quasi-identity matrix corresponding to A; this provides a unique maximal (tangible) solution to supertropical vector equations, via a version of Cramer’s rule. We also describe various properties of this tangible adjoint, and use it to compute supertropical eigenvectors, thereby producing an example in which an n × n matrix has n distinct supertropical eigenvalues but their supertropical eigenvectors are tropically dependent.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Akian, R. Bapat and S. Gaubertt, Max-Plus Algebra, in Handbook of Linear Algebra (L. Hogben, R. Brualdi, A. Greenbaum, R. Mathias, eds.), Chapman and Hall, London, 2006.
M. Akian, S. Gaubert and A. Guterman, Linear independence over tropical semirings and beyond, in The Proceedings of the International Conference on Tropical and Idempotent Mathematics (G. L. Litvinov, S. N. Sergeev, eds.), Contemporary Mathematics 495, American Mathematical Society, Providence, RI, 2009, pp. 1–38. Preprint at arXiv:math.AC/0812.3496v1.
Z. Izhakian, Tropical arithmetic and algebra of tropical matrices, Communications in Algebra 37 (2009), 1445–1468. Preprint at arXiv:math.AG/0505458.
Z. Izhakian, The tropical rank of a tropical matrix, Preprint at arXiv:math.AC/0604208.
Z. Izhakian and L. Rowen, Supertropical algebra, Advances in Mathematics 225 (2010), 2222–2286. Preprint at arXiv:0806.1175.
Z. Izhakian and L. Rowen, Supertropical matrix algebra, Israel Journal of Mathematics 182 (2011), 383–424.
Z. Izhakian and L. Rowen, The tropical rank of a tropical matrix, Communications in Algebra 37 (2009), 3912–3927.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported by the Chateaubriand scientific post-doctorate fellowship, Ministry of Science, French Government, 2007–2008. This research is supported in part by the Israel Science Foundation, grant No. 448/09.
Rights and permissions
About this article
Cite this article
Izhakian, Z., Rowen, L. Supertropical matrix algebra II: Solving tropical equations. Isr. J. Math. 186, 69–96 (2011). https://doi.org/10.1007/s11856-011-0133-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-011-0133-2