# ELT Linear Algebra II

## TL;DR

This paper extends tropical algebra with layered structures, enabling classical algebraic results like Cayley-Hamilton theorem to be adapted to ELT algebra, and introduces new concepts such as the essential trace.

## Contribution

It develops an ELT version of the transfer principle, proves the Cayley-Hamilton theorem in ELT algebra, and introduces the essential trace concept.

## Key findings

- ELT version of the transfer principle proved.
- Cayley-Hamilton theorem established for ELT algebra.
- Introduction and analysis of the essential trace.

## Abstract

This paper is a continuation of [arXiv:1603.02204]. Exploded layered tropical (ELT) algebra is an extension of tropical algebra with a structure of layers. These layers allow us to use classical algebraic results in order to easily prove analogous tropical results. Specifically we prove and use an ELT version of the transfer principal presented in [2]. In this paper we use the transfer principle to prove an ELT version of Cayley-Hamilton Theorem, and study the multiplicity of the ELT determinant, ELT adjoint matrices and quasi-invertible matrices. We also define a new notion of trace -- the essential trace -- and study its properties.