## Abstract

Given a square matrix

*Α*over the integers, we consider the*Z-*module*M*generated by the set of all matrices that are permutation-similar to_{A}*A*. Motivated by analogous problems on signed graph decompositions and block designs, we are interested in the completely symmetric matrices*aI*+*bJ*belonging to*M*. We give a relatively fast method to compute a generator for such matrices, avoiding the need for a very large canonical form over_{A}*Z*. We consider several special cases in detail. In particular, the problem for symmetric matrices answers a question of Cameron and Cioabǎ on determining the eventual period for integers λ such that the λ-fold complete graph λK_{n}has an edge-decomposition into a given (multi)graph.Original language | English |
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Number of pages | 23 |

Publication status | Published - 3 Jan 2022 |