Supertropical linear algebra

Zur Izhakian, Manfred Knebusch and Louis Rowen

Vol. 266 (2013), No. 1, 43–75

DOI: 10.2140/pjm.2013.266.43

Abstract

The objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of “ghost surpasses”. Special attention is paid to the various notions of “base”, which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas the number of elements in various d-bases may differ, it is shown that when an s-base exists, it is unique up to permutation and multiplication by scalars, and can be identified with a set of “critical” elements. Then we turn to orthogonality of vectors, which leads to supertropical bilinear forms and a supertropical version of the Gram matrix, including its connection to linear dependence. We also obtain a supertropical version of a theorem of Artin, which says that if g-orthogonality is a symmetric relation, then the underlying bilinear form is (supertropically) symmetric.

Keywords

tropical algebra, supertropical vector spaces, linear algebra, change of base semirings, linear and bilinear forms, Gram matrix

Mathematical Subject Classification 2010

Primary: 11D09, 15A03, 15A04, 15A15, 15A63, 16Y60

Secondary: 51M20, 20M18, 15A33, 14T05

Milestones

Received: 4 August 2010

Revised: 20 November 2012

Accepted: 20 November 2012

Published: 23 September 2013