The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>A</mml:mi><mml:mi>X</mml:mi><mml:mi>B</mml:mi><mml:mo>=</mml:mo><mml:mi>C</mml:mi></mml:math>

Liu, Xin; Wang, Qingwen

doi:10.1155/2017/9756035

Mathematical Problems in Engineering

2017

DOI: 10.1155/2017/9756035

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The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation AXB=C

Xin Liu

Qingwen Wang

Abstract: For a given generalized reflection matrix , that is, 퐻 = , 2 = , where 퐻 is the conjugate transpose matrix of , a matrix ∈ 푛×푛 is called a Hermitian (anti)reflexive matrix with respect to if 퐻 = and = ± . By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation = over complex field.

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Cited by 2 publications

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Research on Least‐Square η‐Hermitian Solutions of Split Quaternion Matrix Equations

Zhang,

Zhang

2024

Journal of Mathematics

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In this article, we use the real representation matrix of the split quaternion matrix, vector operator, Kronecker product, and Moore–Penrose generalized inverse. We establish the least norm expression of the least‐square η‐Hermitian solution and the least norm expression of the least‐square η‐anti‐Hermitian solution on the split quaternion of the matrix equation AXB + CXD = E. The final solution expression is represented only by real matrices and real vectors. In the algorithm, only real number operations are involved, which avoids complex quaternion operations and greatly reduces the amount of computation. Finally, we use two examples to verify the effectiveness of the proposed algorithm.

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Research on Least‐Square η‐Hermitian Solutions of Split Quaternion Matrix Equations

Zhang,

Zhang

2024

Journal of Mathematics

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A real representation method for special least squares solutions of the quaternion matrix equation $ (AXB, DXE) = (C, F) $

Zhang¹,

Liu²,

Zhao

2022

MATH

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<abstract><p>In this article, our interest is the quaternion matrix equation $ (AXB, DXE) = (C, F) $, and we study its minimal norm centrohermitian least squares solution and skew centrohermitian least squares solution. By applying of the real representation matrices of quaternion matrices and relative properties, we convert the quaternion least squares problems with constrained variables into the corresponding real least squares problems with free variables, and then we obtain the solutions of corresponding problems. The final results can be expressed only by real matrices and vectors, and thus the corresponding algorithms only involves real operations and avoid complex quaternion operations. Therefore, they are portable and convenient. In the end, we give two examples to verify the effectiveness of the purposed algorithms.</p></abstract>

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The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation AXB=C

Cited by 2 publications

References 28 publications

Research on Least‐Square η‐Hermitian Solutions of Split Quaternion Matrix Equations

Research on Least‐Square η‐Hermitian Solutions of Split Quaternion Matrix Equations

A real representation method for special least squares solutions of the quaternion matrix equation $ (AXB, DXE) = (C, F) $

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