RATIONAL SOLUTIONS OF THE MATRIX EQUATION p(X) = A

Abstract

We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA, 258:187-194, 1997. Let p(λ) be any polynomial over ℚ, and let A ∈ Mn(ℚ) have irreducible characteristic polynomial f(λ) with degree n. We provide necessary and sufficient conditions for the existence of a solution X ∈ Mn(ℚ) of the polynomial matrix equation p(X) = A. Specifically, we find necessary and sufficient conditions for f(p(λ)) to have a factor of degree n over ℚ.