On matrix structures invariant under Toda-like isospectral flows. (English) Zbl 0870.15005
A linear transformation \(\tau\) (Toda-like) on the space \(\mathbb{R}^{n\times n}\) of \(n\times n\) real matrices which maps symmetric matrices to antisymmetric ones is introduced. \(\mathbb{R}^{n\times n}\) together with the bilinear operation \(\alpha\), defined by \(\alpha(X,Y)=[X,\tau Y]+[Y,\tau X]\), (where \([U,V]=UV-VU\)) constitute an algebra (usually nonassociative). The paper is mostly devoted to classify a class of such algebras (Toda-like) which arises in connection with eigenvalue computations. The motivation of the paper is to calculate the eigenvalues of large structured symmetric matrices.
[Authors’ remark: “This paper is a somewhat condensed version of a report that we wrote in 1995”].
[Authors’ remark: “This paper is a somewhat condensed version of a report that we wrote in 1995”].
Reviewer: N.D.Sengupta (Bombay)
MSC:
| 15A30 | Algebraic systems of matrices |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 90C05 | Linear programming |
| 15A04 | Linear transformations, semilinear transformations |
Keywords:
Toda-like isospectral flows; Toda algebra; associative and non-associative algebra; Lie bracket; difference-weighted graphs; linear transformation; eigenvalues; large structured symmetric matricesReferences:
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