On the matrix equation XH=HX and the associated controllability problem

Ramadan, Mohamed A.; El-Sayed, Ehab A.

doi:10.1016/j.amc.2006.08.063

Cited by 14 publications

(9 citation statements)

References 7 publications

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“…which satisfy the system of matrix equations A 1 V 1 − E 1 V 1 F 1 = B 1 W 1 and A 2 V 2 − E 2 V 2 F 2 = B 2 W 2 . In this regard, one may use the current work for more understanding of the numerical solution of the Burgers' equation using septic B‐splines, methods based on a septic non‐polynomial spline function for the solution of sixth‐order two‐point boundary value problems and on different matrix equations …”

Section: Illustrative Numerical Examplesmentioning

confidence: 98%

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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Section: Illustrative Numerical Examplesmentioning

confidence: 98%

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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“…Matrices A, E, B ∈ R n×n , R ∈ R n×p and F ∈ R p×p . Output: Matrices V and W. Assumptions: det(E) ≠ 0det(F) ≠ 0, det(B) ≠ 0, and X are nonsingular matrix as shown in [19] and eigenvalues of matrix F must be distinct.…”

Section: Inputmentioning

confidence: 99%

Algorithms for Solving Nonhomogeneous Generalized Sylvester Matrix Equations

El-Sayed

Behady

2020

Mathematical Problems in Engineering

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This paper considers a new method to solve the first-order and second-order nonhomogeneous generalized Sylvester matrix equations AV+BW= EVF+R and MVF2+DV F+KV=BW+R, respectively, where A,E,M,D,K,B, and F are the arbitrary real known matrices and V and W are the matrices to be determined. An explicit solution for these equations is proposed, based on the orthogonal reduction of the matrix F to an upper Hessenberg form H. The technique is very simple and does not require the eigenvalues of matrix F to be known. The proposed method is illustrated by numerical examples.

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“…When dealing with the problems of reordering eigenvalues of regular matrix pairs (Kgström 1993), or computing an additive decomposition of a generalised transform matrix equations (Kgström and Van Dooren 1992;, one naturally encounters the coupled Sylvester matrix equations (1.7). Therefore, the literature on the study of solving matrix equations is large and growing (Ramadan and El-Sayed 2007;Dehghan and Hajarian 2009a,b, 2010a,b, 2011a,b, 2012aZhou, Duan, and Li 2009). In Duan (1996), complete parametric solutions for the generalised Sylvester matrix equation (1.4) have been presented.…”

Section: Introductionmentioning

confidence: 99%

Finite algorithms for solving the coupled Sylvester-conjugate matrix equations over reflexive and Hermitian reflexive matrices

Hajarian

2013

International Journal of Systems Science

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It is known that solving coupled matrix equations with complex matrices can be very difficult and it is sufficiently complicated. In this work, we propose two iterative algorithms based on the Conjugate Gradient method (CG) for finding the reflexive and Hermitian reflexive solutions of the coupled Sylvester-conjugate matrix equations (including Sylvester and Lyapunov matrix equations as special cases). The iterative algorithms can automatically judge the solvability of the matrix equations over the reflexive and Hermitian reflexive matrices, respectively. When the matrix equations are consistent over reflexive and Hermitian reflexive matrices, for any initial reflexive and Hermitian reflexive matrices, the iterative algorithms can obtain reflexive and Hermitian reflexive solutions within a finite number of iterations in the absence of roundoff errors, respectively. Finally, two numerical examples are presented to illustrate the proposed algorithms.

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On the matrix equation XH=HX and the associated controllability problem

Cited by 14 publications

References 7 publications

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

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

Algorithms for Solving Nonhomogeneous Generalized Sylvester Matrix Equations

Finite algorithms for solving the coupled Sylvester-conjugate matrix equations over reflexive and Hermitian reflexive matrices

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