  In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric
Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field
$\mathbf{F}$, any infinite sequence $M_1,M_2,...$ of (skew) symmetric matrices
over $\mathbf{F}$ of bounded $\mathbf{F}$-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 $\sigma$-symmetric matrices
introduced by Rao and myself in [The Rank-Width of Edge-Coloured Graphs,
arXiv:0709.1433v4]. (Skew) symmetric matrices are special cases of
$\sigma$-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 $\sigma$-symmetric matrix form a
delta-matroid. We extend in this way the notion of representability of
delta-matroids by Bouchet.
