Root vectors and Jordan chains of integer matrices (Q1112132)

Let p be a prime and let \(v_ 0,v_ 1,...,v_{k-1}\) be n-vectors with integer components such that \(0\leq v_ i\leq pe\), \(e=(1,1,...,1)^ T\). For a nonsingular \(n\times n\) matrix G with integer entries the sequence \(v_ 0,v_ 1,...,v_{k-1}\) is said to be a Jordan chain associated to p if \(G(v_ 0+v_ 1p+...+v_{k-1}p^{k-1})=p^ kd\) holds for some vector d of integers; the chain is maximal if there is no \(v_ k\) such that \(v_ 0,...,v_{k-1}\), \(v_ k\) is a Jordan chain. The author gives different characterizations of Jordan chains and discusses their role in representations of \(G^{-1}\) of the following form: \(G^{-1}=C(P- N)^{-1}B+M\) where C, B and M are integer matrices and P-N is a Jordan matrix with primes in the main diagonal.