On matrix structures invariant under Toda-like isospectral flows Original Research Article

We work in the space imagen×n of n-by-n real matrices. We say that a linear transformation τ on this space is Toda-like if it maps symmetric matrices to skew-symmetric matrices. With such a transformation we associate a bilinear operation α defined by α(X, Y) colon, equals [X, τY] + [Y, τX] where [U, V] colon, equals UV − VU. Then imagen×n together with this operation is a (usually not associative) algebra. We call any subalgebra of such an algebra a Toda-like algebra. We classify these algebras, which arise in connection with eigenvalue computations. We determine the Toda-like algebras which are of primary interest for this application. We accomplish this classification by studying certain difference-weighted graphs which we associate with the algebras.