Archimedean classes of matrices over ordered fields

TL;DR

This paper introduces a new classification of matrices over ordered fields based on archimedean equivalence, providing canonical forms, lattice structures, and applications to noncommutative geometry and valuation theory.

Contribution

It characterizes archimedean classes of matrices over ordered fields, establishes canonical forms, and explores their algebraic and geometric structures.

Findings

Two matrices are in the same archimedean class iff related by a bounded invertible matrix.

Every class contains a unique row echelon form with determined shape and pivot classes.

A canonical representative exists for matrices over fields of formal Laurent series.

Abstract

Let $(F,\le)$ be an ordered field and let $A,B$ be square matrices over $F$ of the same size. We say that $A$ and $B$ belong to the same archimedean class if there exists an integer $r$ such that the matrices $r A^T A-B^T B$ and $r B^T B-A^T A$ are positive semidefinite with respect to $\le$. We show that this is true if and only if $A=CB$ for some invertible matrix $C$ such that all entries of $C$ and $C^{-1}$ are bounded by some integer. We also show that every archimedean class contains a row echelon form and that its shape and archimedean classes (in $F$) of its pivots are uniquely determined. For matrices over fields of formal Laurent series we construct a canonical representative in each archimedean class. The set of all archimedean classes is shown to have a natural lattice structure while the semigroup structure does not come from matrix multiplication. Our motivation comes from noncommutative real algebraic geometry and noncommutative valuation theory.