Perron-Frobenius theory over real closed fields and fractional power series expansions Original Research Article

Some of the main results of the Perron-Frobenius theory of square nonnegative 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 eigenvalue and normalized eigenvector of real, square, nonnegative, irreducible matrices which are obtained by perturbing a (possibly reducible) nonnegative matrix. Further, we identify a system of equations and inequalities whose solution yields the coefficients of these expansions. For irreducible matrices, our analysis assures that any solution of this system yields a fractional power series with a positive radius of convergence.