Exact rational solution of the matrix equation [formula omitted] by linearization

Abstract

For any given complex n × n matrix A and any polynomial p with complex coefficients, methods to obtain all complex n × n matrix solutions X of A = p ( X ) have been discussed from as early as 1906: however, in practice the “solutions” obtained are only approximations (i.e. 2 n 2 truncated decimal expansions for the real and imaginary parts of the n 2 entries of X). The present article treats the corresponding Diophantine problem where both A and p are defined over the rational field Q , and where, if rational solutions X exist, they are to be found exactly. A complete solution is given when A has no repeated eigenvalue, in which case all rational solutions X are obtained using only linear procedures and integer arithmetic. The method generalizes at once from Q to any finite algebraic extension of Q (or of any Z p ).