Abstract
Due to idempotency of addition, unlike in linear algebra, two different maxpolynomials may be identical as functions. This makes the theory of maxpolynomials in max-algebra different from that in linear algebra. This is reflected in the analogue of the fundamental theorem of algebra, showing that every maxpolynomial can be factorized to linear factors in near linear time. Maxpolynomial equations can also be solved with significantly less effort than in linear algebra.
On the other hand the concept of the characteristic maxpolynomial of a matrix is more tricky, as there are several reasonable, nonequivalent definitions of this concept. The version studied in this chapter is based on the max-algebraic permanent (assignment problem). It is proved that its greatest corner is the principal eigenvalue. Although it is not clear whether all terms of a characteristic maxpolynomial can be found efficiently, it is shown how to find all essential terms in low-order polynomial time.
The max-algebraic Cayley-Hamilton theorem is presented. It has a two-sided form, reflecting the lack of subtraction. It is not easy to find it for a given matrix in general, however, a number of easily solvable special cases is studied.
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Butkovič, P. (2010). Maxpolynomials. The Characteristic Maxpolynomial. In: Max-linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-1-84996-299-5_5
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DOI: https://doi.org/10.1007/978-1-84996-299-5_5
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