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Chebyshevskii Sbornik, 2016, Volume 17, Issue 2, Pages 196–205 (Mi cheb489)  

This article is cited in 1 scientific paper (total in 1 paper)

On the solution of the bilinear matrix equation

S. M. Chuiko

Donbass State Pedagogical University
Full-text PDF (582 kB) Citations (1)
References:
Abstract: Lyapunov matrix equations and their generalizations — linear matrix Sylvester equation widely used in the theory of stability of motion, control theory, as well as the solution of differential Riccati and Bernoulli equations, partial differential equations and signal processing. If the structure of the general solution of the homogeneous part of the Lyapunov equation is well studied, the solution of the inhomogeneous equation Sylvester and, in particular, the Lyapunov equation is quite cumbersome. By using the theory of generalized inverse operators, A. A. Boichuk and S. A. Krivosheya establish a criterion of the solvability of the Lyapunov-type matrix equations $AX - XB = D$ and $X - AXB = D$ and investigate the structure of the set of their solutions. The article A.A. Boichuk and S.A. Krivosheya based on pseudo-inverse linear matrix operator $\mathcal{L},$ corresponding to the homogeneous part of the Lyapunov type equation.
Using the technique of Moore–Penrose pseudo inverse matrices, we suggest an algorithm for finding a family of linearly independent solutions of the bilinear matrix equation and, in particular, the Sylvester matrix equation in general case when the linear matrix operator $\mathcal{L},$ corresponding to the homogeneous part of the bilinear matrix equation, has no inverse. We find an expression for family of linearly independent solutions of the bilinear matrix equation and, in particular, the Sylvester matrix equation in terms of projectors and Moore-Penrose pseudo inverse matrices. This result is a generalization of the result article A. A. Boichuk and S. A. Krivosheya to the case of bilinear matrix equation.
The suggested the solvability conditions and formula for constructing a particular solution of the inhomogeneous bilinear matrix equation is illustrated by an examples.
Bibliography: 17 titles.
Keywords: matrix Sylvester equation, matrix Lyapunov equation, pseudo inverse matrices.
Funding agency Grant number
State Fund for Fundamental Researches (Ukraine) 0109U000381
Received: 02.03.2015
Accepted: 10.06.2016
Bibliographic databases:
Document Type: Article
UDC: 517.9
MSC: 15A24, 34B15, 34C25
Language: Russian
Citation: S. M. Chuiko, “On the solution of the bilinear matrix equation”, Chebyshevskii Sb., 17:2 (2016), 196–205
Citation in format AMSBIB
\Bibitem{Chu16}
\by S.~M.~Chuiko
\paper On the solution of the bilinear matrix equation
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 2
\pages 196--205
\mathnet{http://mi.mathnet.ru/cheb489}
\elib{https://elibrary.ru/item.asp?id=26254434}
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  • https://www.mathnet.ru/eng/cheb/v17/i2/p196
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Abstract page:462
    Full-text PDF :159
    References:79
     
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