Scaled Toda like Flows

This paper discusses the class of isospectral ows X X A X where denotes the Hadamard product and is the Lie bracket The presence of A allows arbitrary and independent scaling for each element in the matrix X The time mapping of the scaled Toda like ow still enjoys a QR like iteration The scaled structure includes the classical Toda ow Brockett s double bracket ow and other interesting ows as special cases Convergence proof is thus uni ed and simpli ed The e ect of scaling on a variety of applications is demonstrated by examples Introduction For simplicity we will con ne our discussion in this paper to the real case only It is convenient to introduce two special subsets in R n S n fX R jX Xg O n fQ R jQQ Ig Recent research has revealed a number of remarkable connections between smooth ows and discrete numerical algorithms Among these a by now classic result is the relationship between the Toda lattice and the QR algorithm That is the time mapping fX k g of the solution X t to the initial value problem X X X X X corresponds exactly to the sequence by applying the QR algorithm to the matrix e In X X X T where X denotes the strictly lower triangular matrix of X The Toda ow latter was generalized to the class X X PL X X X where PL X denotes the projection of X onto a certain speci ed linear subspace L of R n By specifying di erent L gives rise to di erent types of matrix factorizations many of which are in abstract forms For example the following theorem which includes the well known Schur decomposition theorem as a special case has been proved in by using Theorem Given a symmetric matrix A R n there exists a real and orthogonal matrix Q such that the symmetric matrix T QAQ has zero entries in any prescribed positions of T except possibly along the diagonal Another interesting isospectral ow is the so called Brockett s double bracket ow X X X D X X where X and D are matrices in S n and D is xed The ow originally arises as a gradient ow Remarkably it is noticed in that if X is tridiagonal and D diagfn g then coincides precisely with A gradient ow hence becomes Hamiltonian Equation is a special case of a more general projected gradient ows X X X PA X X X where PA X denotes the projection of X S n onto an a ne subspace A of S n The vector eld in represents the projection of the negative gradient of the objective function