TY - JOUR

T1 - Supertropical matrix algebra

AU - Izhakian, Zur

AU - Rowen, Louis

N1 - 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).

PY - 2011/3

Y1 - 2011/3

N2 - 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.•

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.•

UR - http://www.scopus.com/inward/record.url?scp=79953653517&partnerID=8YFLogxK

U2 - 10.1007/s11856-011-0036-2

DO - 10.1007/s11856-011-0036-2

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AN - SCOPUS:79953653517

SN - 0021-2172

VL - 182

SP - 383

EP - 424

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -