ACI-matrices of constant rank over arbitrary fields

Abstract

The columns of an m×n ACI-matrix over a field F are independent affine subspaces of Fm. An ACI-matrix has constant rank ρ if all its completions have rank ρ. Huang and Zhan (2011) [4] characterized the m×n ACI-matrices of constant rank when |F|≥min⁡{m,n+1}. We complete their result characterizing the m×n ACI-matrices of constant rank over arbitrary fields. Quinlan and McTigue (2014) [8] proved that every partial matrix of constant rank ρ has a ρ×ρ submatrix of constant rank ρ if and only |F|≥ρ. We obtain an analogous result for ACI-matrices over arbitrary fields by introducing the concept of complete irreducibility.