Mathematical Proceedings of the Royal Irish Academy

Abstract:

Sometime in the early 1970s Trevor introduced me to the spectral theory of positive linear operators which owes its origins to the celebrated Perron-Frobenius theorem according to which the spectral radius of a non-negative matrix is one of its eigenvalues, and possesses a corresponding eigenvector whose components are nonnegative real numbers. This was a subject dear to his heart, and a recurring theme to which he often returned in later years. In this article we make a tangential contribution to the converse of the Perron-Frobenius theorem, the so-called inverse eigenvalue problem for non-negative matrices, namely, under what circumstances are the components of a vector of complex numbers the eigenvalues of such a matrix? To this end, we associate with each vector of unit norm an analytic self map of the unit open disc of the complex plane, which is also a rational function, and develop its power series expansion about the origin. Sufficient conditions are presented that ensure that the resulting coefficients—which encode information about the chosen vector—are non-negative. Conversely, if these are all non-negative, it turns out that the vector satisfies conditions that are necessary ones for it to solve the inverse problem.