Abstract
It is shown that the polynomial matrix equation AT=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=AT=p(A) or A=p(A) where p(0)>0 Finally, equations AT=p(A) and A=p(A) where p(0)=0 are reduced to equations AT=aAk and A=aAk supplemented by some divisibility conditions on the exponents occurring in p(λ). The solutions of these monomial equations have been characterized earlier.
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© 1994 Springer-Verlag Berlin Heidelberg
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Turakainen, P. (1994). On polynomial matrix equations XT=p(X) and X=p(X) Where all parameters are nonnegative. In: Karhumäki, J., Maurer, H., Rozenberg, G. (eds) Results and Trends in Theoretical Computer Science. Lecture Notes in Computer Science, vol 812. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58131-6_63
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DOI: https://doi.org/10.1007/3-540-58131-6_63
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