This paper continues the series of publications devoted to surveying and developing methods for solving the following problems for a two-parameter matrix F (λ, μ) of general form: exhausting points of the mixed regular spectrum of F (λ, μ); performing operations on polynomials in two variables (computing the GCD and LCM of a few polynomials, division of polynomials, and factorization); computing a minimal basis of the null-space of polynomial solutions of the matrix F (λ, μ) and separation of its regular kernel; inversion and pseudo in version of polynomial and rational matrices in two variables, and solution of systems of nonlinear algebraic equations in two unknowns. Most of the methods suggested are based on rank factorizations of a two-parameter polynomial matrix and on the method of hereditary pencils. Bibliography: 8 titles.
Similar content being viewed by others
References
V. N. Kublanovskaya, “To solving problems of algebra for two-parameter matrices. 1,” Zap. Nauchn. Semin. POMI, 359, 107–149 (2008).
V. N. Kublanovskaya, “To solving problems of algebra for two-parameter matrices. 2,” Zap. Nauchn. Semin. POMI, 359, 150–165 (2008).
V. N. Kublanovskaya and V. B. Khazanov, “To solving problems of algebra for two-parameter matri es. 3,” Zap. Nauchn. Semin. POMI, 359, 166–207 (2008).
V. N. Kublanovskaya, “To solving problems of algebra for two-parameter matrices. 4,” Zap. Nauchn. Semin. POMI, 367, 121–144 (2009).
V. N. Kublanovskaya and V. B. Khazanov, Numerical Methods for Solving Parametric Problems of Algebra. Part 1. One-Parameter Problems [in Russian], Nauka, St. Petersburg (2004).
V. N. Kublanovskaya, “Methods and algorithms for solving spectral problems for polynomial and rational matrices,” Zap. Nauchn. Semin. POMI, 238 ,3–33 (1997).
V. N. Kublanovskaya and V. N. Simonova, “An approach to solving systems of nonlinear algebraic equations,” Zap. Nauchn. Semin. POMI, 202, 71–96 (1992).
V. N. Kublanovskaya and V. B. Khazanov, “To solving problems of algebra for two-parameter matrices. 6,” Zap. Nauchn. Semin. POMI, 367, 171–186 (2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 367, 2009, pp. 145–171.
Rights and permissions
About this article
Cite this article
Kublanovskaya, V.N. To solving problems of algebra for two-parameter matrices. V. J Math Sci 165, 574–588 (2010). https://doi.org/10.1007/s10958-010-9827-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-010-9827-y