ACI-matrices of constant rank over arbitrary fields

2016/04/12 by Borobia, Alberto, Canogar, Roberto
#15A54 #FOS: Mathematics #Rings and Algebras (math.RA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1604.03447

Abstract

The columns of a m× n ACI-matrix over a field \mathbbF are independent affine subspaces of \mathbbF^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 |\mathbbF|≥ 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 |\mathbbF|≥ ρ. We obtain an analogous result for ACI-matrices over arbitrary fields by introducing the concept of complete irreducibility.

Related

• On affine spaces of alternating matrices with constant rank
• A dimension bound for constant rank subspaces of matrices over a finite field
• On affine spaces of rectangular matrices with constant rank
• The classification of large spaces of matrices with bounded rank
• On the cardinality of matrices with prescribed rank and partial trace over a finite field