  We consider different phase spaces for the Toda flows and the less familiar
SVD flows. 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 transform 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.
