Perron - Frobenlus Theory Over Real Closed Fields and Fractional Power

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 'Research was supported in part by National Science Foundation grants DMS-92-07409, DMS-88-0144S and DMS-91-2331B and by United States-Israel Binational Science Foundation grant 90-00434. LINEAR ALGEBRA AND ITS APPLICATIONS 220:123-150 (995) © Elsevier Science Inc., 1995 655 Avenue of the Americas, New York, NY 10010 0024-3795/95/$9.50 SSDI 0024-3795(94)OOO53-G 124 B. C. EAVES, U. G. ROTHBLUM, AND H. SCHNEIDER 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.