  We discuss a number of properties of CMV matrices, by which we mean the class
of unitary matrices recently introduced by Cantero, Moral, and Velazquez. 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 behaviour of
this system.
