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Special Matrices

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CiteScore 2018: 0.64

SCImago Journal Rank (SJR) 2018: 0.408
Source Normalized Impact per Paper (SNIP) 2018: 1.005

Mathematical Citation Quotient (MCQ) 2018: 0.29

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2300-7451
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Polynomial sequences generated by infinite Hessenberg matrices

Luis Verde-Star
  • Department of Mathematics, Universidad Autónoma Metropolitana, Iztapalapa, Apartado 55-534, México D. F. 09340, Mexico
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Published Online: 2017-01-20 | DOI: https://doi.org/10.1515/spma-2017-0002

Abstract

We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.

Keywords: Polynomial sequences of interpolatory type; infinite Toeplitz matrices; infinite Hessenberg matrices; Toeplitz companion matrices

References

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  • [2] A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Springer, New York, 1999.Google Scholar

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  • [4] M. Merca, A note on the determinant of a Toeplitz-Hessenberg matrix, Spec. Matrices, 2013; 1:10-16.Google Scholar

  • [5] L. Verde-Star, Infinite triangular matrices, q-Pascal matrices, and determinantal representations, Linear Algebra Appl. 434 (2011) 307-318.Google Scholar

  • [6] L. Verde-Star, Characterization and construction of classical orthogonal polynomials using a matrix approach, Linear Algebra Appl. 438 (2013) 3635-3648.Google Scholar

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  • [8] L. Verde-Star, Elementary triangular matrices and inverses of k-Hessenberg matrices and triangular matrices, Spec. Matrices, 2015; 3: 250-256.Google Scholar

About the article

Received: 2016-08-29

Accepted: 2016-09-26

Published Online: 2017-01-20

Published in Print: 2017-01-26


Citation Information: Special Matrices, Volume 5, Issue 1, Pages 64–72, ISSN (Online) 2300-7451, DOI: https://doi.org/10.1515/spma-2017-0002.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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