In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field F, any infinite sequence M_1,M_2,... of (skew) symmetric matrices over F of bounded F-rank-width has a pair i< j, such that M_i is isomorphic to a principal submatrix of a principal pivot transform of M_j. We generalise this result to σ-symmetric matrices introduced by Rao and myself in [The Rank-Width of Edge-Coloured Graphs, arXiv:0709.1433v4]. (Skew)

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... metric matrices are special cases of σ-symmetric matrices. As a by-product, we obtain that for every infinite sequence G_1,G_2,... of directed graphs of bounded rank-width there exist a pair i<j such that G_i is a pivot-minor of G_j. Another consequence is that non-singular principal submatrices of a σ-symmetric matrix form a delta-matroid. We extend in this way the notion of representability of delta-matroids by Bouchet.