  We analyze the correspondence between finite sequences of finitely supported
probability distributions and finite-dimensional, real, symmetric, tridiagonal
matrices. In particular, we give an intrinsic description of the topology
induced on sequences of distributions by the usual Euclidean structure on
matrices. Our results provide an analytical tool with which to study ensembles
of tridiagonal matrices, important in certain inverse problems and integrable
systems. As an application, we prove that the Euler characteristic of any
generic isospectral set of symmetric, tridiagonal matrices is a tangent number.
