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

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