In this paper, we mainly investigate the general solution and its minimum norm of the dual quaternion generalized Sylvester matrix equation <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$AX-YB=C$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mo>-</mo> <mi>Y</mi> <mi>B</mi> <mo>=</mo> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation>. For the general solution, we derive it via three different orders, and the mutual transformation relations between these three sets of general solutions are investigated by unifying the constant terms. On the other hand, our previous research has revealed that the minimum F-norm solution of a matrix equation over the dual quaternion algebra may not exist, and even if it exists, it may not be unique. Therefore, we introduce the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$F^+$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>F</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>-norm in this paper to address this problem. It is worth noting that the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$F^+$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>F</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>-norm lacks homogeneity of a norm, but it satisfies the properties of paranorms. On this basis, we explore the minimum <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$F^+$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>F</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>-norm of the aforementioned general solution. Furthermore, since <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$AX-YB=C$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mo>-</mo> <mi>Y</mi> <mi>B</mi> <mo>=</mo> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation> is a bivariate equation, we also derive the necessary and sufficient conditions for the simultaneous minimization of the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$F^+$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>F</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>-norms of the general solution (X, Y). Finally, two numerical examples are presented to illustrate the main results of this paper.

General solution and minimum $F^+$-norm solution of the generalized Sylvester matrix equation over the dual quaternion algebra

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发表日期：

2026年6月20日

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Ling-Jie Zhu,
Qing-Wen Wang

摘要

In this paper, we mainly investigate the general solution and its minimum norm of the dual quaternion generalized Sylvester matrix equation $AX-YB=C$ $A X - Y B = C$ . For the general solution, we derive it via three different orders, and the mutual transformation relations between these three sets of general solutions are investigated by unifying the constant terms. On the other hand, our previous research has revealed that the minimum F-norm solution of a matrix equation over the dual quaternion algebra may not exist, and even if it exists, it may not be unique. Therefore, we introduce the $F^+$ $F^{+}$ -norm in this paper to address this problem. It is worth noting that the $F^+$ $F^{+}$ -norm lacks homogeneity of a norm, but it satisfies the properties of paranorms. On this basis, we explore the minimum $F^+$ $F^{+}$ -norm of the aforementioned general solution. Furthermore, since $AX-YB=C$ $A X - Y B = C$ is a bivariate equation, we also derive the necessary and sufficient conditions for the simultaneous minimization of the $F^+$ $F^{+}$ -norms of the general solution (X, Y). Finally, two numerical examples are presented to illustrate the main results of this paper.

General solution and minimum \(F^+\)-norm solution of the generalized Sylvester matrix equation over the dual quaternion algebra

摘要