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In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or SymmetricMatrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field$\mathbf{F}$, any infinite sequence $M_1,M_2,...$ of (skew) symmetric matricesover $\mathbf{F}$ of bounded $\mathbf{F}$-rank-width has a pair $i< j$, suchthat $M_i$ is isomorphic to a principal submatrix of a principal pivottransform of $M_j$. We generalise this result to $\sigma$-symmetric matricesintroduced 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 infinitesequence $G_1,G_2,...$ of directed graphs of bounded rank-width there exist apair $i<j$ such that $G_i$ is a pivot-minor of $G_j$. Another consequence isthat non-singular principal submatrices of a $\sigma$-symmetric matrix form adelta-matroid. We extend in this way the notion of representability ofdelta-matroids by Bouchet.

Well-Quasi-Ordering of Matrices under Schur Complement and Applications to Directed Graphs