<p>In this paper, we first propose the real representation matrix of the complex split quaternion matrix by using the semi-tensor product. Then, applying its properties, the MP inverse, the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varvec{vec}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">vec</mi> </mrow> </math></EquationSource> </InlineEquation> operator and the Kronecker product of real matrices, we give the expressions of the minimal norm least squares solution, pure imaginary least squares solution and complex least squares solution of the complex split quaternion matrix equations <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varvec{AX=B}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">A</mi> <mi mathvariant="bold-italic">X</mi> <mo mathvariant="bold">=</mo> <mi mathvariant="bold-italic">B</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varvec{AXC=B}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">A</mi> <mi mathvariant="bold-italic">X</mi> <mi mathvariant="bold-italic">C</mi> <mo mathvariant="bold">=</mo> <mi mathvariant="bold-italic">B</mi> </mrow> </math></EquationSource> </InlineEquation>. Subsequently, the corresponding real algorithms of calculating the above least squares solutions are proposed, and their effectiveness are demonstrated by numerical examples. Finally, we deal with the color image restoration problem by applying the minimal norm pure imaginary least squares solution of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\varvec{AX=B}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">A</mi> <mi mathvariant="bold-italic">X</mi> <mo mathvariant="bold">=</mo> <mi mathvariant="bold-italic">B</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Real algorithms for special least squares solutions of the complex split quaternion matrix equations $AX=B$ and $AXC=B$

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发表日期：

2025年6月24日

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Zhihan Zhou,
Zixiang Meng,
Yuqing Zhang,
Ying Song,
Fengxia Zhang

摘要

In this paper, we first propose the real representation matrix of the complex split quaternion matrix by using the semi-tensor product. Then, applying its properties, the MP inverse, the $\varvec{vec}$ $vec$ operator and the Kronecker product of real matrices, we give the expressions of the minimal norm least squares solution, pure imaginary least squares solution and complex least squares solution of the complex split quaternion matrix equations $\varvec{AX=B}$ $A X = B$ and $\varvec{AXC=B}$ $A X C = B$ . Subsequently, the corresponding real algorithms of calculating the above least squares solutions are proposed, and their effectiveness are demonstrated by numerical examples. Finally, we deal with the color image restoration problem by applying the minimal norm pure imaginary least squares solution of $\varvec{AX=B}$ $A X = B$ .

Real algorithms for special least squares solutions of the complex split quaternion matrix equations \(AX=B\) and \(AXC=B\)

摘要