Supertropical matrix algebra

被引：0

作者：

Zur Izhakian

Louis Rowen

机构：

[1] Bar-Ilan University,Department of Mathematics

[2] CNRS et Universite Denis Diderot (Paris 7),undefined

来源：

Israel Journal of Mathematics | 2011年 / 182卷

关键词：

Simple Cycle; Vandermonde Matrix; Idempotent Semiring; Rank Defect; Tangible Element;

D O I：

暂无

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摘要：

The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows: The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible.There exists an adjoint matrix adj(A) such that the matrix A adj(A) behaves much like the identity matrix (times |A|).Every matrix A is a supertropical root of its Hamilton-Cayley polynomial fA. If these roots are distinct, then A is conjugate (in a certain supertropical sense) to a diagonal matrix.The tropical determinant of a matrix A is a ghost iff the rows of A are tropically dependent, iff the columns of A are tropically dependent.Every root of fA is a “supertropical” eigenvalue of A (appropriately defined), and has a tangible supertropical eigenvector.

引用

共 6 条

[1]

Birkhoff G.(1946)Tres observaciones sobre el algebra lineal Universidad Nacional de Tucuman Revista, Serie A 5 147-151

[2]

Izhakian Z.(2009)Tropical arithmetic and algebra of tropical matrices Communications in Algebra 37 1445-1468

[3]

Izhakian Z.(2010)Supertropical algebra Advances in Mathematics 225 2222-2286

[4]

Rowen L.(2009)The tropical rank of a matrix Communications in Algebra 37 3912-3927

[5]

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[6]

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