Linear independence over tropical semirings and beyond

By Marianne Akian, Stephane Gaubert, Alexander E. Guterman. 2008

157 citations recorded by the source corpus.

We investigate different notions of linear independence and of ma- trix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a no- tion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/b9bfe2f00a229a66e7b7a90d35f375a0fde4da88

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Citations in the corpus

Supertropical linear algebra (2010, background): “2006] and linear dependence [Akian et al. 2009], the emphasis often is combinatoric or geometric.”
Max-linear Systems: Theory and Algorithms (2010, background): “This vector is also an eigenvector for 2 f4; 5g and thus (A;B) = [4; 5] :”
Tropical Cramer Determinants Revisited (2013, methodology): “Some of the following notions were introduced in [AGG09], where details and additional properties can be found.”