Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices (extended abstract)

We prove that every infinite sequence of skew-symmetric or symmetric matrices M 1 , M 2 , … over a fixed finite field must have a pair M i , M j ( i < j ) such that M i is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in M j , if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymourʼs theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittleʼs theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oumʼs theorem for graphs of bounded rank-width with respect to pivot-minors.