Maximal Generalized Rank in Graphical Matrix Spaces

TL;DR

This paper extends a combinatorial characterization of maximal bounded rank subspaces in graphical matrix spaces to generalized rank functions and to alternating matrix spaces for general graphs.

Contribution

It introduces two significant extensions of existing rank characterization results to broader classes of rank functions and matrix spaces.

Findings

Characterization valid for generalized rank functions including permanental rank

Extension to bounded rank subspaces of graphical alternating matrix spaces

Applicable to a wide class of bipartite and general graphs

Abstract

In this note we prove two extensions of a recent combinatorial characterization due to Li, Qiao, Wigderson, Wigderson and Zhang (arXiv:2206.04815) of the maximal dimension of bounded rank subspaces of the graphical matrix space associated with a bipartite graph. Our first result shows that the above characterization remains valid for a wide class of generalized rank functions, including e.g. the permanental rank. Our second result extends the characterization to bounded rank subspaces of the graphical alternating matrix space associated with a general graph.