L-free directed bipartite graphs and echelon-type canonical forms

Abstract

It is common knowledge that matrices can be brought in echelon form by Gaussian elimination and that the reduced echelon form of a matrix is canonical in the sense that it is unique. In [4], working within the context of the algebra \( \mathbb{C}^{n\times n}_{\mathrm{upper}}\) of upper triangular n×n matrices, certain new canonical forms of echelon-type have been introduced. Subalgebras of \( \mathbb{C}^{n\times n}_{\mathrm{upper}}\) determined by a pattern of zeros have been considered too. The issue there is whether or not those subalgebras are echelon compatible in the sense that the new canonical forms belong to the subalgebras in question. In general they don’t, but affirmative answers were obtained under certain conditions on the given zero pattern. In the present paper these conditions are weakened. Even to the extent that a further relaxation is not possible because the conditions involved are not only sufficient but also necessary. The results are used to study equivalence classes in \( \mathbb{C}^{m\times n}\) associated with zero patterns. The analysis of the pattern of zeros referred to above is done in terms of graph theoretical notions.