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We consider different phase spaces for the Toda flows and the less familiarSVD flows. For the Toda flow, we handle symmetric and non-symmetric matriceswith real simple eigenvalues, possibly with a given profile. Profiles encode,for example, band matrices and Hessenberg matrices. For the SVD flow, we assumesimplicity 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 anddiagonal matrix. A variant construction transform the flows into uniformstraight line motion. Since limit points belong to the phase space, asymptoticbehavior becomes a local issue. The constructions rely only on basic facts oflinear algebra, making no use of symplectic geometry.

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