## Supertropical matrix algebra. III: Powers of matrices and their supertropical eigenvalues.(English)Zbl 1283.15055

Summary: We investigate powers of supertropical matrices, with special attention to the role of the coefficients of the supertropical characteristic polynomial (especially the supertropical trace) in controlling the rank of a power of a matrix. This leads to a Jordan-type decomposition of supertropical matrices, together with a supertropical eigenspace decomposition of a power of an arbitrary supertropical matrix.
For Parts I and II, see [Isr. J. Math. 182, 383–424 (2011; Zbl 1215.15018); ibid. 186, 69–96 (2011; Zbl 1277.15013)].

### MSC:

 15A30 Algebraic systems of matrices 15A15 Determinants, permanents, traces, other special matrix functions 15A80 Max-plus and related algebras 16S50 Endomorphism rings; matrix rings

### Citations:

Zbl 1215.15018; Zbl 1277.15013
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### References:

 [1] Akian, M.; Bapat, R.; Gaubert, S., MAX-plus algebra, () [2] Akian, M.; Gaubert, S.; Guterman, A., Linear independence over tropical semirings and beyond, (), 1-38 · Zbl 1182.15002 [3] Akian, M.; Gaubert, S.; Walsh, C., Discrete MAX-plus spectral theory, (), 19-51 [4] Cahen, G.; Dubois, D.; Quadrat, J.-P.; Viot, M., A linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing, IEEE trans. automat. control, AC-30, 2, 10-220, (1985) · Zbl 0557.93005 [5] Izhakian, Z., Tropical arithmetic and tropical matrix algebra, Comm. algebra, 37, 4, 1445-1468, (2009) · Zbl 1165.15017 [6] Izhakian, Z.; Knebusch, M.; Rowen, L., Supertropical linear algebra, (2010), preprint · Zbl 1240.13003 [7] Izhakian, Z.; Rowen, L., Supertropical algebra, Adv. math., 225, 4, 2222-2286, (2010) · Zbl 1273.14132 [8] Izhakian, Z.; Rowen, L., The tropical rank of a tropical matrix., Comm. algebra, 37, 11, 3912-3927, (2009) · Zbl 1184.15003 [9] Izhakian, Z.; Rowen, L., Supertropical matrix algebra, Israel J. math., 182, 1, 383-424, (2011) · Zbl 1215.15018 [10] Z. Izhakian, L. Rowen, Supertropical matrix algebra II: solving tropical equations, Israel J. Math. (2009), in press; preprint, arXiv:0902.2159. · Zbl 1277.15013 [11] Izhakian, Z.; Rowen, L., Supertropical resultants, J. algebra, 324, 8, 1860-1886, (2010) · Zbl 1227.12008 [12] Straubing, H., A combinatorial proof of the Cayley-Hamilton theorem, Discrete math., 43, 2-3, 273-279, (1983) · Zbl 0533.15010
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