ON MATRIX STRUCTURES INVARIANT UNDER TODA-LIKE ISOSPECTRAL FLOWS

We work in the space R(n x n) of n-by-n real matrices. We say that a l inear transformation tau on this space is Toda-like if it maps symmetr ic matrices to skew-symmetric matrices. With such a transformation we associate a bilinear operation alpha defined by alpha(X, Y) := [X, tau Y] + [Y, tau X] where [U, V] := UV - VU. Then R(n x n) together with this operation is a (usually not associative) algebra. We call any sub algebra of such an algebra a Toda-like algebra. We classify these alge bras, which arise in connection with eigenvalue computations. We deter mine the Toda-like algebras which are of primary interest for this app lication. We accomplish this classification by studying certain differ ence-weighted graphs which we associate with the algebras. (C) Elsevie r Science Inc., 1997.