CMV: The unitary analogue of Jacobi matrices

Abstract

We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices studied recently by Cantero, Moral, and Velázquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behavior of this system.

An Introduction to Random Matrices

“…In this language, the key observation of [KiN04] is that the Verblunsky coefficients corresponding to Haar-distributed unitaries are independent. See also [FoR06], [KiN07] and [BoNR08] for further developments in this direction.…”

CMV matrices in random matrix theory and integrable systems: a survey

“…Percy Deift showed us a derivation of the Jacobi matrix result using the symplectic structure naturally associated to the Toda lattice. The same can be done in the circular case (see [KilNen2,Corollary 8.6]). The key idea is the following: As we can write the underlying symplectic form in either set of variables, we can view (4.9) as a symplectomorphism between two concrete symplectic manifolds.…”

A note on Poisson brackets for orthogonal polynomials on the unit circle

“…Their purpose was to understand the complete integrability of the defocusing Ablowitz-Ladik (AL) equation with periodic boundary conditions; in the process, they compute certain fairly complicated combinations of the Poisson brackets of the Wall polynomials-see Proposition 2.9. (For the origin of the Ablowitz-Ladik equation, see [1], [2]; for recent results on this integrable system, obtained through its connection to OPUC, see for example [3,4,7,10,11], or the review paper [13].) In [3], Cantero and Simon found all but one of the Poisson brackets for the monic orthogonal and the second kind polynomials.…”

“…The first proof of this formula (see [4] for some of its very interesting consequences) was done in [4] using the r-matrix formulation of the Ablowitz-Ladik bracket (see also [7] and [9] for more details on this alternate way of defining the Poisson bracket (1.3)).…”