Congruence of rational matrices defined by an integer matrix

Abstract

We study algorithms that construct an invertible matrix B∈Mn(Z) that defines congruence Btr·X·B=Y of given square rational matrices X,Y∈Mn(Q). We describe a general algorithm and discuss special cases of positive-definite and singular matrices. In the first case, the algorithm solves the decision problem and unequivocally determines if such an integer matrix exists.Since the problem of isomorphism of various finite structures can be viewed as a special case of matrix congruence problem, our results are applicable in this setting. We discuss this topic and show that the presented algorithms solve this problem in the case of partially ordered sets, vertex and edge labeled directed and undirected graphs. Moreover, they allow computing the automorphism group Aut(G) of a given (di)graph G.The results of this work are applicable in the general framework of Coxeter spectral analysis of signed graphs Δ introduced in (SIAM J. Discrete Math. 27:827–854, 2013) as they make it possible to calculate strong and weak Gram Z-congruences. Moreover, they can be used to compute isotropy groups and thus are useful in the study of matrix morsifications in the sense of (J. Pure Appl. Algebra 215:3–34, 2011).