PERCA},ION \onlinear Anall sis 47 12001 ) 1905-19 I 7 lfonlinear Analysis rvrvw. elsevier.nl/locate/na Entropy of Square Non-negative Matrices M. X. He Noua Southeastern Uniuersi,ty, Flori'da, USA P. E. Ricci Llniuersitd degli, Studi, di, Roma "La Sap'ienza", Italg D. Simon Noua Southeastern Un'iuersity, Flori'da, USA Abstract Let A be an r x r adjacency matrix associated with a graph G and Xa the shift spaces. The entropy of shift space X4 associated with the matrix A with non-negative integer elements is defined in 15] as h(xA) :.,gg; log lB"(Xa)1, is the number of n-blocks appearing in points of X, and the zeta /x , \ (o{f )-.*P (I o'l'' "l' \;1 n/ where p"(rp) is the number of periodic points of period n of a dynamicalsystem (LI,Q).In this paper we extend the entropy and the zeta function to the square matrix A t'ith non-negative real elements. We take the sum of the i.l-th entry of the n-th power of matrix A. S,(A): !iA")i,; to define the entropy of the matrix A. X,J h(A): "i*;1oeS,(A) and the trace of matrix A, Tn(A) : Tr(-4') to define the zeta function ed(t) : exp (tf-1 '"f)*). The recurrence reiation of the entropy sequences {S"(A)}Er is obtained and zeta function is explicitly determined. Furthermore we compute the entropy and zeta, function of some important special matrices. Keg utords: Entropy, Zeta function, Powet of matrices 0362-546X/01/$ - see lront matter O 2001 Published by Elsevier Science Ltd. PII: S0362-546X(0 I )00320-0 where lB"(Xa)l function as