The CMV-matrix and the generalized Lanczos process

The CMV-matrix is the five-diagonal matrix that represents the operator of multiplication by an independent variable in a special basis formed of Laurent polynomials orthogonal on the unit circle $C$. The article by Cantero, Moral, and Velázquez, which was published in 2003 and described this matrix, has attracted much attention because it implied that the conventional orthogonal polynomials on $C$ can be interpreted as the characteristic polynomials of the leading principal submatrices of a certain five-diagonal matrix. In this publication, we remind about the fact that finite-dimensional sections of the CMV-matrix emerged in papers on the unitary eigenvalue problem long before the article by Cantero et al. Moreover, band forms were also found for a number of other situations in the normal eigenvalue problem.