Linearizing Toda and SVD flows on large phase spaces of matrices with real spectrum

Abstract

We consider different phase spaces for the Toda flows (Flaschka, 1974; Moser, 1975) and the less familiar SVD flows (Chu, 1986; Li, 1997). For the Toda flow, we handle symmetric and non-symmetric matrices with real simple eigenvalues, possibly with a given profile. Profiles encode, for example, band matrices and Hessenberg matrices. For the SVD flow, we assume simplicity of the singular values. In all cases, an open cover is constructed, as are corresponding charts to Euclidean space. The charts linearize the flows, converting it into a linear differential system with constant coefficients and diagonal matrix. A variant construction transforms the flows into uniform straight line motion. Since limit points belong to the phase space, asymptotic behavior becomes a local issue. The constructions rely only on basic facts of linear algebra, making no use of symplectic geometry.