ACI-matrices of constant rank over arbitrary fields

TL;DR

This paper extends the characterization of ACI-matrices with constant rank to all fields, introducing the concept of complete irreducibility to generalize previous results.

Contribution

It completes the classification of constant rank ACI-matrices over arbitrary fields and introduces the concept of complete irreducibility for these matrices.

Findings

Extended Huang and Zhan's characterization to arbitrary fields.

Established a new result on submatrices of constant rank over any field.

Introduced the concept of complete irreducibility for ACI-matrices.

Abstract

The columns of a $m\times n$ ACI-matrix over a field $\mathbb{F}$ are independent affine subspaces of $\mathbb{F}^m$. An ACI-matrix has constant rank $\rho$ if all its completions have rank $\rho$. Huang and Zhan (2011) characterized the $m\times n$ ACI-matrices of constant rank when $|\mathbb{F}|\geq \min\{m,n+1\}$. We complete their result characterizing the $m\times n$ ACI-matrices of constant rank over arbitrary fields. Quinlan and McTigue (2014) proved that every partial matrix of constant rank $\rho$ has a $\rho\times \rho$ submatrix of constant rank $\rho$ if and only $|\mathbb{F}|\geq \rho$. We obtain an analogous result for ACI-matrices over arbitrary fields by introducing the concept of complete irreducibility.