Supertropical matrix algebra

Zur Izhakian; Louis Rowen

doi:10.1007/s11856-011-0036-2

Supertropical matrix algebra

Zur Izhakian, Louis Rowen

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

42 Scopus citations

Abstract

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 {divides}A{divides}).• Every matrix A is a supertropical root of its Hamilton-Cayley polynomial f_A. 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 f_A is a "supertropical" eigenvalue of A (appropriately defined), and has a tangible supertropical eigenvector.•

Original language	English
Pages (from-to)	383-424
Number of pages	42
Journal	Israel Journal of Mathematics
Volume	182
Issue number	1
DOIs	https://doi.org/10.1007/s11856-011-0036-2
State	Published - Mar 2011

Bibliographical note

Funding Information:
The first author is supported by the Chateaubriand scientific post-doctorate fellowship, Ministry of Science, French Government, 2007-2008 This research is supported by the Israel Science Foundation (grants No. 1178/06 and 448/09).

Funding

The first author is supported by the Chateaubriand scientific post-doctorate fellowship, Ministry of Science, French Government, 2007-2008 This research is supported by the Israel Science Foundation (grants No. 1178/06 and 448/09).

Funders	Funder number
French government
Israel Science Foundation	448/09, 1178/06
Ministry of science and technology, Israel

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10.1007/s11856-011-0036-2

Cite this

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abstract = "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 {divides}A{divides}).• 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.•",

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AB - 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 {divides}A{divides}).• 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.•

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Supertropical matrix algebra

Abstract

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