The relation between the Jordan structure of a matrix and its graph (Q2365692)

The height characteristic of a matrix \(A\) is the sequence \(n(A^ i)- n(A^{i-1})\) where \(n\) is dimension of the null space. It is proved that this strongly majorizes a sequence obtained from the graph of \(A\) by considering decompositions into paths, and which of the paths extend to cycles of length one greater. This was previously established by \textit{H. Schneider} and the author for triangular matrices [Path coverings of graphs and height characteristics of matrices. J. Combin. Theory, Ser. B (to appear)].