Well-quasi-ordering of matrices under Schur complement and applications to directed graphs

In [S. Oum, Rank-width and well-quasi-ordering of skew-symmetric or symmetric matrices, Linear Algebra and its Applications 436 (7) (2012) 2008–2036] Oum proved that, for a fixed finite field FF, any infinite sequence M1,M2,…M1,M2,… of (skew) symmetric matrices over FF of bounded FF-rank-width   has a pair i