Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices

TL;DR

This paper establishes a well-quasi-ordering property for infinite sequences of symmetric or skew-symmetric matrices over a finite field with bounded rank-width, extending similar results from graph and matroid theory.

Contribution

It generalizes existing theorems on well-quasi-ordering from graphs and matroids to matrices with bounded rank-width, unifying these concepts under a broader framework.

Findings

Proves a new well-quasi-ordering theorem for matrices over finite fields.

Extends classical graph and matroid theorems to matrix settings.

Demonstrates the applicability of rank-width in matrix isomorphism problems.

Abstract

We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in M_j, if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour's theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum's theorem for graphs of bounded rank-width with respect to pivot-minors.