On polynomial matrix equations X^T=p(X) and X=p(X) Where all parameters are nonnegative

Abstract

It is shown that the polynomial matrix equation A^T=p(A) does not have any nonnegative nonsymmetric solution if the coefficients of p(λ) are nonnegative and the constant term p(0) is positive. The equation is studied in [4] where some necessary conditions for the existence of such solutions are presented. Then a structural characterization is given for nonnegative square matrices A such that A=A^T=p(A) or A=p(A) where p(0)>0 Finally, equations A^T=p(A) and A=p(A) where p(0)=0 are reduced to equations A^T=aA^k and A=aA^k supplemented by some divisibility conditions on the exponents occurring in p(λ). The solutions of these monomial equations have been characterized earlier.