Abstract
Linear equations and operators in a space of matrices are investigated. The transformations of matrix equations which allow one to find the conditions of solvability and the inertial properties of Hermite solutions are determined. New families of matrices (collectives) are used in the theory of inertia and positive invertibility of linear operators and, in particular, in the problems of localization of matrix spectra and matrix beams.
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References
F. R. Gantmakher,Matrix Theory [in Russian], Nauka, Moscow (1989).
Kh. D. Ikaramov,Numerical Solution of Matrix Equations [in Russian], Nauka, Moscow (1984).
D. G. Korenevskii,Stability of Dynamical System under Random Perturbations of Parameters. Algebraic Criteria [in Russian], Naukova Dumka, Kiev (1989).
A. G. Mazko,Matrix Equations and Collectives [in Russian], Preprint, Institute of Mathematics, Ukrainian Academy of Sciences, No. 83, Kiev (1989).
A. G. Mazko, “Semiinversion and properties of matrix invariants,”Ukr. Mat. Zh.,40, No 4, 525–528 (1988).
R. Horn and Ch. Johnson,Matrix Analysis [Russian translation], Mir, Moscow (1989).
A. G. Mazko,Theory of Distribution of Matrix Spectrum with Respect to Algebraic and Transcendental Curves [in Russian], Preprint, Institute of Mathematics, Ukrainian Academy of Sciences, No. 5, Kiev (1983).
F. Chojnowski and S. Gutman, “Root-clustering criteria (II); linear matrix equations,”IMA J. Math. Contr. & Inf.,6, 289–300 (1989).
R. D. Hill, “Inertia theory for simultaneously triangulable complex matrices,”Linear Algebra Its Appl.,2, 131–142 (1969).
M. A. Krasnoselskii, E. A. Lifshits, and A. V. Sobolev,Positive Linear Systems [in Russian], Nauka, Moscow (1985).
Yu. L. Daletskii and M. G. Krein,Stability of Solutions to Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).
A. G. Mazko, “Criterion for matrix spectrum to belong to an arbitrary region in a certain class,”Avtomatika, No. 6, 54–59 (1980).
A. G. Mazko, “On the problem of distribution of the spectrum of a regular matrix beam,” in:Direct Methods in Problems of Dynamics and Stability of Multidimensional Systems [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1986), pp. 99–110.
A. G. Mazko, “Distribution of spectrum of a regular matrix beam with respect to plane curves,”Ukr. Mat. Zh.,38, No. 1, 116–120 (1986).
A. G. Mazko, “Generalized Lyapunov equation for a regular matrix beam,” in:Mathematical Methods for Investigation of Applied Problems in the Dynamics of Bodies Carrying Liquid [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1992), pp. 67–76.
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 60–68, January, 1993.
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Mazko, A.G. Transformations and inertia of solutions to linear matrix equations. Ukr Math J 45, 66–75 (1993). https://doi.org/10.1007/BF01062039
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DOI: https://doi.org/10.1007/BF01062039