Isospectral Flows

Abstract

The Toda flow is a dynamical system whose dependent variables may be viewed as the entries of a symmetric tridiagonal matrix. The spectrum of the matrix is invariant in time and, as $t \to \infty $, the off-diagonal entries tend to zero, exposing the eigenvalues on the main diagonal. Recently a wave of interest in the Toda and related isospectral flows was sparked in the numerical analysis community by the suggestion that to integrate the Toda flow numerically might be a cost effective way to calculate the eigenvalues of large symmetric tridiagonal matrices. A second source of interest was the recently discovered connection between the Toda flow and the $QR$ algorithm. The present paper has two principal aims. The first is to point out that the Toda and related flows are intimately connected with the power method. This connection clarifies completely the convergence properties of the flows. The relationship of the Toda flow to both the $QR$ algorithm and the power method is based on a connection with the $QR$ matrix factorization. The second aim of this paper is to introduce and discuss families of isospectral flows associated with two other well-known matrix factorizations, the $LU$ factorization and the Cholesky factorization.