Centrohermitian and Skew-Centrohermitian Solutions to the Minimum Residual and Matrix Nearness Problems of the Quaternion Matrix Equation $${(AXB,DXE) = (C,F)}$$ ( A X B , D X E ) = ( C , F )

Şimşek, S.; Sarduvan, Murat; Özdemir, Halim

doi:10.1007/s00006-016-0688-4

Cited by 19 publications

(3 citation statements)

References 18 publications

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“…According to Theorem 3.2, (13), and (18), Problem 1 has a solution X ∈ H E if and only if (19) holds. From ( 17) and Theorem 3.2, ( 14) implies (20).…”

Section: Methods I For the Solution Of Problemmentioning

confidence: 99%

“…Linear matrix equation plays an important role in physics, computer science, data analysis, engineering, signal and color image processing, control and system theory, optimal control, neural network, stability theory, optimal control, and so on. Many researchers pay attention to studying the solutions of matrix equations such as the Lyapunov equation

A^{H} X + X A = B, A X = B, A X B = C, A X B + C Y D = E

, and

A X B + C X D = E

over the real field, complex field, and the quaternion skew field (see previous studies 1‐30 and references cited therein). It is worthy to note that η ‐Hermitian quaternion matrices are applied to widely linear modeling for the quaternion involution properties and have a great influence upon convergence analysis in statistical signal processing (see previous studies 31‐34 for details).…”

Section: Introductionmentioning

confidence: 99%

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Direct methods onη‐Hermitian solutions of the split quaternion matrix equation (AXB,CXD)=(E,F)

Yuan

Jiang

2021

Math Methods in App Sciences

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This paper provides two direct methods for solving the split quaternion matrix equation where X is an unknown split quaternion η‐Hermitian matrix, and A, B, C, D, E, F are known split quaternion matrices with suitable size. Our tools are the Kronecker product, Moore–Penrose generalized inverse, real representation, and complex representation of split quaternion matrices. Our main work is to find the necessary and sufficient conditions for the existence of a solution of the matrix equation mentioned above, derive the explicit solution representation, and provide four numerical algorithms and two numerical examples.

show abstract

“…According to Theorem 3.2, (13), and (18), Problem 1 has a solution X ∈ H E if and only if (19) holds. From ( 17) and Theorem 3.2, ( 14) implies (20).…”

Section: Methods I For the Solution Of Problemmentioning

confidence: 99%

A^{H} X + X A = B, A X = B, A X B = C, A X B + C Y D = E

, and

A X B + C X D = E

Section: Introductionmentioning

confidence: 99%

Direct methods onη‐Hermitian solutions of the split quaternion matrix equation (AXB,CXD)=(E,F)

Yuan

Jiang

2021

Math Methods in App Sciences

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“…Due to these wide ranges of applications for Sylvester equations, as well as quaternions, the problem of obtaining solutions to (1.1), specifically over the skew-field of quaternions, has attracted considerable attention [12][13][14][15]. Matrix equations other than Sylvester equations over quaternions have also been studied in the literature [16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

A Global Krylov Subspace Method for the Sylvester Quaternion Matrix Equation

Şimşek

2024

Journal of New Theory

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This study concerns the Sylvester matrix equation in the quaternion setting when the coefficient matrices as well as the unknown matrix have quaternion entries. We propose a global Generalized Minimal Residual (GMRES) method for the solution of such a matrix equation. The proposed approach works directly with the Sylvester operator to generate orthonormal bases for Krylov subspaces formed of matrices. Then, the best approximate matrix solution to the Sylvester equation at hand in such a Krylov subspace is constructed from a matrix minimizing the Frobenius norm of the residual. We describe how this minimization of the residual norm can be carried out efficiently and report numerical results on real examples related to image restoration.

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Algebraic techniques for the least squares problems in elliptic complex matrix theory and their applications

Kösal

Pekyaman

2021

Math Methods in App Sciences

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In this study, we introduce concepts of norms of elliptic complex matrices and derive the least squares solution, the pure imaginary least squares solution, and the pure real least squares solution with the least norm for the elliptic complex matrix equation AX = B by using the real representation of elliptic complex matrices. To prove the authenticity of our results and to distinguish them from existing ones, some illustrative examples are also given. Elliptic numbers are generalized form of complex and so real numbers. Thus, the obtained results extend, generalize, and complement some known least squares solutions results from the literature. KEYWORDS elliptic complex matrices, elliptic complex numbers, least squares solution with the minimum norm, real representation of elliptic complex matrices MSC CLASSIFICATION 30G35; 15A09How to cite this article: Kösal HH, Pekyaman M. Algebraic techniques for the least squares problems in elliptic complex matrix theory and their applications.

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Centrohermitian and Skew-Centrohermitian Solutions to the Minimum Residual and Matrix Nearness Problems of the Quaternion Matrix Equation $${(AXB,DXE) = (C,F)}$$ ( A X B , D X E ) = ( C , F )

Cited by 19 publications

References 18 publications

Direct methods onη‐Hermitian solutions of the split quaternion matrix equation (AXB,CXD)=(E,F)

Direct methods onη‐Hermitian solutions of the split quaternion matrix equation (AXB,CXD)=(E,F)

A Global Krylov Subspace Method for the Sylvester Quaternion Matrix Equation

Algebraic techniques for the least squares problems in elliptic complex matrix theory and their applications

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