Linear independence over tropical semirings and beyond Export

@article{citeulike:3890389, abstract = {We investigate different notions of linear independence and of matrix 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 notion 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.}, archivePrefix = {arXiv}, author = {Akian, Marianne and Gaubert, Stephane and Guterman, Alexander}, citeulike-article-id = {3890389}, citeulike-linkout-0 = {http://arxiv.org/abs/0812.3496}, citeulike-linkout-1 = {http://arxiv.org/pdf/0812.3496}, day = {18}, eprint = {0812.3496}, keywords = {tropical}, month = {Dec}, posted-at = {2009-01-16 08:58:16}, priority = {2}, title = {Linear independence over tropical semirings and beyond}, url = {http://arxiv.org/abs/0812.3496}, year = {2008} }

TY - JOUR ID - citeulike:3890389 L3 - citeulike-article-id:3890389 N2 - We investigate different notions of linear independence and of matrix 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 notion 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. TI - Linear independence over tropical semirings and beyond KW - tropical AU - Akian, Marianne AU - Gaubert, Stephane AU - Guterman, Alexander PY - 2008/Dec/18/ UR - http://arxiv.org/abs/0812.3496 ER -