PERRON-FROBENIUS THEORY OVER REAL CLOSED FIELDS AND FRACTIONAL POWER-SERIES EXPANSIONS

Some of the main results of the Perron-Frobenius theory of square nonn egative matrices over the reals are extended to matrices with elements in a real closed field. We use the results to prove the existence of a fractional power series expansion for the Perron-Frobenius eigenvalu e and normalized eigenvector of real, square, nonnegative, irreducible matrices which are obtained by perturbing a (possibly reducible) nonn egative matrix. Further, we identify a system of equations and inequal ities whose solution yields the coefficients of these expansions. For irreducible matrices, our analysis assures that any solution of this s ystem yields a fractional power series with a positive radius of conve rgence.