Fractional Minimal Rank

TL;DR

This paper introduces the fractional minimal rank for partial matrices, determines it for matrices with bipartite graphs as minimal cycles, and explores related minimal rank problems with open questions.

Contribution

It defines fractional minimal rank and computes it for matrices with bipartite graphs as minimal cycles, advancing understanding of partial matrix ranks.

Findings

Fractional minimal rank lies between triangular and minimal rank.

Determined fractional minimal rank for matrices with bipartite minimal cycle graphs.

Identified minimal rank of partial block matrices with invertible entries on minimal cycles.

Abstract

The notion of fractional minimal rank of a partial matrix is introduced, a quantity that lies between the triangular minimal rank and the minimal rank of a partial matrix. The fractional minimal rank of partial matrices whose bipartite graph is a minimal cycle are determined. Along the way, we determine the minimal rank of a partial block matrix with invertible given entries that lie on a minimal cycle. Some open questions are stated.