Entropy of square non-negative matrices

Let A be an r x r adjacency matrix associated with a graph G and X-A the shift spaces. The entropy of shift space X-A associated with the matrix A with non-negative integer elements is defined in [5] as h(X-A) = lim(n --> infinity) 1/n log/B-n(X-A)/, where /B-n(X-A)/ is the number of n-blocks appearing in points of X, and the zeta function as zeta (phi)(t) = exp (Sigma (infinity)(n=1) Pn (phi)/n t(n)), where p(n)(phi) is the number of periodic points of period n of a dynamicalsystem (M, phi). In this paper we extend the entropy and the zeta function to the square matrix A with non-negative real elements. We take the sum of the ij-th entry of the n-th power of matrix A, S-n(A) = Sigma (r)(i,j)(A (n))(i,j) to define the entropy of the matrix A, h(A) = lim(n)--> (infinity) 1/n log S-n(A) and the trace of matrix A(n), T-n(A) = Tr(A(n)) to define the zeta function zeta (A)(t) = exp (Sigma (infinity)(n=1) T-n(A)/n t(n)). The recurrence relation of the entropy sequences {S-n(A)}(n=1)(infinity) is obtained and zeta function is explicitly determined. Furthermore we compute the entropy and zeta function of some important special matrices.