Supertropical matrix algebra

TL;DR

This paper develops a comprehensive algebraic framework for supertropical matrix algebra, establishing properties of determinants, adjoints, eigenvalues, and matrix roots within this novel mathematical setting.

Contribution

It introduces fundamental theorems and structures for supertropical matrices, extending classical linear algebra concepts into the supertropical domain.

Findings

Tropical determinant is multiplicative for tangible determinants.

Existence of an adjoint matrix with identity-like properties.

Matrices are supertropical roots of their Hamilton-Cayley polynomial.

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.