An atlas for tridiagonal isospectral manifolds

Let TΛ be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix Λ with simple spectrum. We introduce bidiagonal coordinates, charts defined on open dense domains forming an explicit atlas for TΛ. In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in TΛ now lies in the interior of some chart domain. We provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics, both for continuous and discrete time.