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 M1, M2, . . . over a fixed finite field must have a pair Mi, Mj (i < j) such that Mi is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in Mj , 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 Seymour’s theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle’s theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum’s theorem for graphs of bounded rank-width with respect to pivotminors.