Poset block equivalence of integral matrices

Boyle, Mike; Huang, Danrun

doi:10.1090/S0002-9947-03-02947-7

Poset block equivalence of integral matrices
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by Mike Boyle and Danrun Huang PDF

Trans. Amer. Math. Soc. 355 (2003), 3861-3886 Request permission

Abstract:

Given square matrices $B$ and $B’$ with a poset-indexed block structure (for which an $ij$ block is zero unless $i\preceq j$), when are there invertible matrices $U$ and $V$ with this required-zero-block structure such that $UBV = B’$? We give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain $\mathcal R$. As one application, when $\mathcal R$ is a field we classify such matrices up to similarity by matrices respecting the block structure. We also give complete invariants for equivalence under the additional requirement that the diagonal blocks of $U$ and $V$ have determinant $1$. The invariants involve an associated diagram (the “$K$-web”) of $\mathcal R$-module homomorphisms. The study is motivated by applications to symbolic dynamics and $C^*$-algebras.

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