  Let ${\cal T}_\Lambda$ be the compact manifold of real symmetric tridiagonal
matrices conjugate to a given diagonal matrix $\Lambda$ with simple spectrum.
We introduce {\it bidiagonal coordinates}, charts defined on open dense domains
forming an explicit atlas for ${\cal T}_\Lambda$. In contrast to the standard
inverse variables, consisting of eigenvalues and norming constants, every
matrix in ${\cal T}_\Lambda$ 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.
