DIRECTED GRAPHS AND KRONECKER INVARIANTS OF PAIRS OF MATRICES

Call two pairs (M,N) and (M′,N′) of m × n matrices over a field K, simultaneously K-equivalent if there exist square invertible matrices S,T over K, with M′ = SMT and N′ = SNT. Kronecker [2] has given a complete set of invariants for simultaneous equivalence of pairs of matrices. Associate in the natural way to a finite directed graph Γ, with v vertices and e edges, an ordered pair (M,N) of e × v matrices of zeros and ones. It is natural to try to compute the Kronecker invariants of such a pair (M,N), particularly since they clearly furnish isomorphism-invariants of Γ. Let us call two graphs "linearly equivalent" when their two corresponding pairs are simultaneously equivalent. There have existed, since 1890, highly effective algorithms for computing the Kronecker invariants of pairs of matrices of the same size over a given field [1,2,5,6] and in particular for those arising in the manner just described from finite directed graphs. The purpose of the present paper, is to compute directly these Kronecker ...