Abstract
This paper is concerned with tridiagonal matrices of the form ┌ ┐ │g(1) 1 │ │ 1 g(2) 1 │ A(g)=│ . . . │. │ 1 g(n-1) 1 │ │ 1 g(n)│ └ ┘ where g=(g(1),g(2),...,g(n)) is a real vector in Rn. We treat such a matrix as a function of its diagonal vector g=(g(1),g(2),...,g(n)) and investigate, by means of a three-term recurrence relation, the properties of a partition of Rn which consists of "regular domains" and "singular surfaces" with respect to A(g). In particular, properties related to geometrical, topological, symmetry and oscillatory properties of a general partition are given, and analytic estimates of the sizes of the regular domains, as well as existence and localization of eigenvalues related to linear and nonlinear difference eigenvalue problems are derived as application.