Nonlinear eigenvalue { eigenvector problems for STP matrices
نویسنده
چکیده
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, 14]) and by Krein and Gantmacher in the 1930s. 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, 8] is a somewhat expanded version of their 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 de ne for each x 2 R 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. Thus, for example,
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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...
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