An Atlas Adapted to the Toda Flow

Torres, David Martínez; Tomei, Carlos

doi:10.1093/imrn/rnac210

Cited by 2 publications

(1 citation statement)

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“…A first example were bidiagonal variables for Jacobi matrices [12]. An extension [22] considered the manifold of full flags of any non-compact real semisimple Lie algebra and its Hessenberg-type submanifolds. These references and the present self-contained paper make no use of symplectic theory, the constructions using classical tools from linear algebra.…”

Section: Introductionmentioning

confidence: 99%

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

Leite¹,

Saldanha²,

Torres³

et al. 2023

Preprint

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We consider different phase spaces for the Toda flows ([9, 15]) and the less familiar SVD flows ([4, 14]). 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.

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Section: Introductionmentioning

confidence: 99%