2008/06/06 by Zur Izhakian, Izhakian, Zur, Louis Rowen +1
Computer Science · Engineering · Mathematics · #15A03 #15A15 #5A09 #65F15 #Advanced Numerical Analysis Techniques #Advanced Topics in Algebra #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.CO #msc:15A03 #msc:15A15 #msc:5A09 #msc:65F15
paper · pdf · doi:10.48550/arxiv.0806.1178
24 pages
openalex publication_date 2008/06/06 · arxiv created 2009/12/07 · arxiv updated 2009/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \adjA such that the matrix A \adjA 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.