2 P. DEIFT, L. C. LI, AND C. TOMEI M = QR- M' = RQ , (1.5) which is isospectral, as M' = Q~l MQ. In the case of (1.4), M is now a particular quadratic matrix polynomial M = M(X) = EQ + XE1 + X2E2 , (1.6) where the coefficients E{, i = 0,1,2, depend on X and Y in an explicit way (see below). and the role of the QR factorization is played by a particular, unique factorization of M into first order matrix polynomials M(X) = (B0 + JBiA)(C0 + CiA) . (1.7) Exchanging the factors, M'(X) = (Co + d X)(B0 + BXX) (1.8) implements the mapping ^ associated with (1.4). There are, in addition, further conse- quences: (i) The isospectral nature of the map M(X) -• M'(X) implies the 'integrability' of ^ , and (ii) ^ linearizes on the Jacobi variety of the associated curve {(A, 77) £ W2 : det(M(A) 7?) = 0} . Over the last decade, following the seminal work of Symes ([Syl], [Sy2]), the QR algorithm, and related algorithms such as the LU algorithm and the Cholesky algorithm, have come to be understood (see, for example, [Chu], [Wa], [DLNT], [DLT]) as time-one maps of completely integrable Hamiltonian systems closely related with the Toda flow and its generalizations. The underlying symplectic structures are Lie-Poisson structures on the coadjoint orbits of particular Lie-algebras, which in turn are double Lie-algebras carrying classical R-matices satisfying the (modified) Yang-Baxter equation. Recall that j?-matrix theory (see, for example, [STS], [FT] see also [DL] for a more pedestrian account) gives a natural explanation of the existence of many commuting integrals, and also leads in
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