Closed-form solutions to Sylvester-conjugate matrix equations

Wu, Ai‐Guo; Feng, Gang; Duan, Guang‐Ren; Wu, Wei-Jun

doi:10.1016/j.camwa.2010.04.035

Cited by 28 publications

(10 citation statements)

References 47 publications

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“…In [285] a closed-form solution to the equation for f (X) =X is considered, together with the set of all possible solutions for (7.4) and for the bilinear problem AX + BY = XF . Chiang, Duan, Feng, Wu and their collaborators have thoroughly investigated these formulations and their role in control applications.…”

Section: The Stein and Discrete Lyapunov Equations The Stein And Thementioning

confidence: 99%

Computational Methods for Linear Matrix Equations

Simoncini¹

2016

SIAM Rev.

404

434

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Given the matrices A, B, C, D and E of conforming dimensions, we consider the linear matrix equation AXE + DXB = C in the unknown matrix X. Our aim is to provide an overview of the major algorithmic developments that have taken place in the past few decades in the numerical solution of this and of related problems, which are becoming a reliable tool in the numerical formulation of advanced application models.

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Section: The Stein and Discrete Lyapunov Equations The Stein And Thementioning

confidence: 99%

Computational Methods for Linear Matrix Equations

Simoncini¹

2016

SIAM Rev.

404

434

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“…Proposition 1. If (X s , Y s ) is a special solution to the regulator-conjugate matrix equation (6) and (X * , Y * ) is the general solution to the Sylvester-conjugate matrix equation (7), then the general solution to the regulator-conjugate matrix equation (6) is given by…”

Section: Ifmentioning

confidence: 99%

“…The above result can be called as superposition principle in linear regulatorconjugate matrix equation. According to this principle, to solve regulator-conjugate equation (6), one need solve the corresponding homogeneous equation and obtain a special solution of the equation itself. Up to this point, the only thing we should do is to find a special solution to matrix equation (6).…”

Section: Ifmentioning

confidence: 99%

“…with Z ∈ C r×p being an arbitrarily chosen free parameter matrix, satisfy the matrix equation (6). Further, define D(s) as in (5).…”

Section: Remarkmentioning

confidence: 99%

“…For instance, in [4], the necessary and sufficient condition of the existing of solution to complex matrix equations X − AXB = C and X − AXB = C are given by using the tool of real representation, and the problem is converted into solving real matrix equations. The complex matrix equation with the form of AX + BY = XF is investigated in [6], where explicit closed-form solutions are provided in terms of controllability and observability matrices with a significant degree of freedom. In addition, solutions to the matrix equation XF − AX = C have also been deeply investigated ( [9], [3], [2] and so on).…”

mentioning

confidence: 99%

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Parametric solutions to the regulator-conjugate matrix equations

Zhang¹,

Zhu²,

Wu³

et al. 2017

Journal of Industrial &Amp; Management Optimization

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The problem of solving regulator-conjugate matrix equations is considered in this paper. The regulator-conjugate matrix equations are a class of nonhomogeneous equations. Utilizing several complex matrix operations and the concepts of controllability-like and observability-like matrices, a special solution to this problem is constructed, which includes solving an ordinary algebraic matrix. Combined with our recent results on Sylvester-conjugate matrix equations, complete solutions to regulator-conjugate matrix equations can be obtained by superposition principle. The correctness and effectiveness are verified by a numerical example. 2010 Mathematics Subject Classification. 15A24.

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A combination of Sylvester block sum and block matrix Kronecker map for explicit solutions of Sylvester system of matrix equations

Ramadan

Bayoumi

Hadhoud

2019

Math Methods in App Sciences

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In this paper, we consider an explicit solution of system of Sylvester matrix equations of the form being arbitrary matrices, where V 1 ,W 1 ,V 2 and W 2 are the matrices to be determined. First, the definitions, of the matrix polynomial of block matrix, Sylvester sum, and Kronecker product of block matrices are defined. Some definitions, lemmas, and theorems that are needed to propose our method are stated and proved. Numerical test problems are solved to illustrate the suggested technique. KEYWORDS degree of freedom, explicit solutions, Kronecker map, Sylvester matrix equations, Sylvester sum MSC CLASSIFICATION 15A24; 15A69; 15A09

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Closed-form solutions to Sylvester-conjugate matrix equations

Cited by 28 publications

References 47 publications

Computational Methods for Linear Matrix Equations

Computational Methods for Linear Matrix Equations

Parametric solutions to the regulator-conjugate matrix equations

A combination of Sylvester block sum and block matrix Kronecker map for explicit solutions of Sylvester system of matrix equations

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