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CMV: The unitary analogue of Jacobi matrices
Author(s) -
Killip Rowan,
Nenciu Irina
Publication year - 2007
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20160
Subject(s) - mathematics , symplectic geometry , hermitian matrix , integrable system , pure mathematics , unitary state , unitary matrix , circular ensemble , foliation (geology) , jacobi method , algebra over a field , hierarchy , mathematical analysis , market economy , geochemistry , political science , economics , law , metamorphic rock , geology
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. © 2006 Wiley Periodicals, Inc.