Non - Linear Eigenvalue - EigenvectorProblems for STP

We consider the eigenvalue-eigenvector problem where p 1 p m?1 = r. We prove an analogue of the classical Gantmacher{Krein Theorem for the eigenvalue-eigenvector structure of STP matrices in the case where p i 1 for each i, plus various extensions thereof. A matrix A is said to be strictly totally positive (STP) if all its minors are strictly positive. STP matrices were independently introduced by Schoenberg in 1930, see 13] (also to be found in 14]), and by Krein and Gantmacher in the 1930's. The main results concerning eigenvalues and eigenvectors of STP matrices were proved by Gantmacher and Krein in their 1937 paper 6]. (An announcement appeared in 1935 in 5]. Chapter 2 of their book 7] and 8] is a somewhat expanded version of the paper 6].) Among the results proved in that paper is that an N N STP matrix has N positive, simple eigenvalues, and the eigenvector associated with the ith eigenvalue, in descending order of magnitude, has i ? 1 sign changes. To explain this more precisely let us deene for each x 2 IR N two sign change indices. These are S ? (x) which is simply the number of ordered sign changes in the vector x where zero entries are discarded, and S + (x) which is the maximum number of ordered sign changes in the vector x where zero entries are given arbitrary values. Note also that S ? (0) = 0, while for convenience we will set S + (0) = N. We now can formally state the Gantmacher{Krein Theorem.