ACI-matrices of constant rank over arbitrary fields

Abstract

The columns of a m × n ACI-matrix over a field F are independent affine subspaces of F m . An ACI-matrix has constant rank ρ if all its completions have rank ρ. Huang and Zhan (2011) 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) 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. *

The WST-decomposition for partial matrices

“…The Rank of partial matrices has a substantial literature (see Section 1 of [4]). The constantRank partial matrices were studied in [7], and the constantRank ACI-matrices were studied in [1,2,3,4,6].…”