Related papers: CMV matrices: Five years after

The CMV matrices are the unitary analogs of Jacobi matrices. In the finite case, it is well-known that the set of Jacobi matrices with a fixed trace is nothing but a coadjoint orbit of the lower triangular group. In this note, we will give…

We present a survey of recent results concerning a remarkable class of unitary matrices, the CMV matrices. We are particularly interested in the role they play in the theory of random matrices and integrable systems. Throughout the paper we…

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…

We find a finite CMV matrix whose eigenvalues coincide with the Dirichlet data of a circular periodic problem. As a consequence, we obtain circular analogues of the classical trace formulae for periodic Jacobi matrices.

We develop subordinacy theory for extended CMV matrices. That is, we provide explicit supports for the singular and absolutely continuous parts of the canonical spectral measure associated with a given extended CMV matrix in terms of the…

We show that the Jacobi polynomials that are orthogonal on the unit circle (the Jacobi OPUC) are CMV bispectral. This means that the corresponding Laurent polynomials in the CMV basis satisfy two dual ordinary eigenvalue problems: a…

We prove a bijective unitary correspondence between 1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum and 2) special periodic block-CMV matrices satisfying a Magic Formula. This…

We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related with the theory of CMV matrices. It contains an arbitrary parameter which leads to a linear…

First, we give summary of the present values of CKM matrix elements. Then, we discuss whether CKM matrix is unitary or not, and how we can find out if it is not unitary.

The author discusses integrability of Hamiltonian dynamical systems in the aftermath of KdV. The author also discusses the role of integrable systems in certain numerical computations, particularly the computation of the eigenvalues of a…

The CMV matrices and their sub-matrices are applied to the description of all solutions to the Schur interpolation problem for contractive analytic operator-valued functions in the unit disk (the Schur class functions).

We investigate the symmetries of so-called generalized extended CMV matrices. It is well-documented that problems involving reflection symmetries of standard extended CMV matrices can be subtle. We show how to deal with this in an elegant…

We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP…

We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This…

If the unitary quark- mixing matrix, $V$, is moduli symmetric then it depends on three real parameters. This means that there is a relation between the four parameters needed to parametrize a general $V$. It is shown that there exists a…

Two matrices $A$ and $B$ are called unitary (resp. orthogonal) equivalent if $AU=VB$ for two unitary (resp. orthogonal) matrices $U$ and $V$. Using trace identities, criteria are given for simultaneous unitary, orthogonal or complex…

We develop a procedure for determining whether a square complex matrix is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. Our approach has several advantages over existing methods. We discuss these differences and…

For Schrodinger operators, there is a well known and widely used formula connecting the transfer matrices and Dirichlet determinants. No analog of this formula was previously known for CMV matrices. In this paper we fill this gap and…

M. Derevyagin, L. Vinet and A. Zhedanov introduced in Constr. Approx. 36 (2012) 513-535 a new connection between orthogonal polynomials on the unit circle and the real line. It maps any real CMV matrix into a Jacobi one depending on a real…

The CKM matrix, V, relates the quark mass and flavor bases. In the standard model, V is unitary 3X3, and specified by four arbitrary parameters, including a phase allowing for $CP$ violation. We review the experimental determination of V,…