TL;DR
This paper investigates the factorization properties of tropical matrices, proving factorizability for 2x2 cases and identifying conditions for 3x3 matrices, while introducing a universal closure operation that is always factorizable.
Contribution
It establishes the factorizability of all tropically non-singular 2x2 matrices and characterizes factorizability conditions for 3x3 matrices, extending tropical matrix theory.
Findings
All non-singular 2x2 tropical matrices are factorizable.
Certain 3x3 matrices are factorizable based on Bruhat decomposition.
A universal closure operation via the tropical adjoint is always factorizable.
Abstract
In contrast to the situation in classical linear algebra, not every tropically non-singular matrix can be factored into a product of tropical elementary matrices. We do prove the factorizability of any tropically non-singular 2x2 matrix and, relating to the existing Bruhat decomposition, determine which 3x3 matrices are factorizable. Nevertheless, there is a closure operation, obtained by means of the tropical adjoint, which is always factorizable, generalizing the decomposition of the closure operation * of a matrix.
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Taxonomy
TopicsAdvanced Topics in Algebra · Polynomial and algebraic computation · Algebraic structures and combinatorial models