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Supertropical matrix algebra. II: Solving tropical equations. (English) Zbl 1277.15013

Summary: 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 \(n \times n\) matrix has \(n\) distinct supertropical eigenvalues but their supertropical eigenvectors are tropically dependent.
For part I, cf. [Isr. J. Math. 182, 383–424 (2011; Zbl 1215.15018)].

MSC:

15A30 Algebraic systems of matrices
15A15 Determinants, permanents, traces, other special matrix functions
15A18 Eigenvalues, singular values, and eigenvectors
15A80 Max-plus and related algebras
14T05 Tropical geometry (MSC2010)
16S50 Endomorphism rings; matrix rings
16Y60 Semirings

Citations:

Zbl 1215.15018
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References:

[1] 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.
[2] 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. · Zbl 1182.15002
[3] 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
[4] Z. Izhakian, The tropical rank of a tropical matrix, Preprint at arXiv:math.AC/0604208. · Zbl 1184.15003
[5] Z. Izhakian and L. Rowen, Supertropical algebra, Advances in Mathematics 225 (2010), 2222–2286. Preprint at arXiv:0806.1175. · Zbl 1273.14132
[6] Z. Izhakian and L. Rowen, Supertropical matrix algebra, Israel Journal of Mathematics 182 (2011), 383–424. · Zbl 1215.15018
[7] Z. Izhakian and L. Rowen, The tropical rank of a tropical matrix, Communications in Algebra 37 (2009), 3912–3927. · Zbl 1184.15003
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