The CMV matrix is the five-diagonal matrix that represents the operator of multiplication by the independent variable in a special basis formed of Laurent polynomials orthogonal on the unit circle C. The article by Cantero, Moral, and Velázquez, published in 2003 and describing this matrix, has attracted much attention because it implies 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. The present paper recalls that finite-dimensional sections of the CMV matrix appeared in papers on the unitary eigenvalue problem long before the article by Cantero et al. was published. Moreover, band forms were also found for a number of other situations in the normal eigenvalue problem.
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M. J. Cantero, L. Moral, and L. Velázquez, “Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle,” Linear Algebra Appl., 362, 29–56 (2003).
A. Bunse-Gerstner and L. Elsner, “Schur parameter pencils for the solution of the unitary eigenproblem,” Linear Algebra Appl., 154–156, 741–778 (1991).
T. Bella, V. Olshevsky, and P. Zhlobich, “A quasi-separable approach to five-diagonal CMV and Fiedler matrices,” Linear Algebra Appl., 434, 957–976 (2011).
L. Elsner and Kh. D. Ikramov, “On a condensed form for normal matrices under finite sequences of elementary unitary similarities,” Linear Algebra Appl., 254, 79–98 (1997).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 463, 2017, pp. 142–153.
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Ikramov, K.D. The CMV Matrix and the Generalized Lanczos Process. J Math Sci 232, 837–843 (2018). https://doi.org/10.1007/s10958-018-3913-y
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DOI: https://doi.org/10.1007/s10958-018-3913-y