An atlas for tridiagonal isospectral manifolds

Abstract

Let ${\mathcal{T}}_{\Lambda }$ 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 ${\mathcal{T}}_{\Lambda }$. In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in ${\mathcal{T}}_{\Lambda }$ 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.

Fixed typos; 16 pages, 3 figures

References in corpus (3)

• The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence
• Spectral distributions and isospectral sets of tridiagonal matrices
• The asymptotics of Wilkinson's shift iteration

Cited by in corpus (6)

• The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence
• Dynamics of the symmetric eigenvalue problem with shift strategies
• Linearizing Toda and SVD flows on large phase spaces of matrices with real spectrum
• On the open Toda chain with external forcing
• An atlas adapted to the Toda flow
• Convergence rates to deflation of simple shift strategies