Supertropical matrix algebra

Zur Izhakian, Louis Rowen

Research output: Contribution to journalArticle

27 Citations (Scopus)

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 |A|).
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 languageEnglish
Pages (from-to)383-424
Number of pages42
JournalIsrael Journal of Mathematics
Volume182
Issue number1
DOIs
Publication statusPublished - Mar 2011

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Matrix Algebra
Determinant
Roots
Unit matrix
Algebraic Theory
Dependent
Cayley
Diagonal matrix
Eigenvector
Multiplicative
Eigenvalue
Distinct
Polynomial

Cite this

Supertropical matrix algebra. / Izhakian, Zur; Rowen, Louis .

In: Israel Journal of Mathematics, Vol. 182, No. 1, 03.2011, p. 383-424.

Research output: Contribution to journalArticle

Izhakian, Zur ; Rowen, Louis . / Supertropical matrix algebra. In: Israel Journal of Mathematics. 2011 ; Vol. 182, No. 1. pp. 383-424.
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