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<p>Multilevel matrix is a special kind of block matrices, each block has smaller blocks. In this paper, we discuss the necessary and sufficient conditions under which a matrix belongs to a kind of hierarchical structured matrices, and this kind of matrices commute with two <i>c</i>-involutory matrices. The singular value decomposition of such multilevel matrices are discussed. Moreover, we characterize the consistent solutions, which possess the structure described above, of the system <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3234_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="140" /> </InlineMediaObject> <EquationSource Format="TEX">\(AX=B, XC=D\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mo>=</mo> <mi>B</mi> <mo>,</mo> <mi>X</mi> <mi>C</mi> <mo>=</mo> <mi>D</mi> </mrow> </math></EquationSource> </InlineEquation> under a kernel space constraint. A matrix nearness problem of these consistent solutions has been considered. A numerical algorithm and some numerical examples are presented to demonstrate the effectiveness and feasibility of our results.</p>

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A matrix nearness problem constrained on the kernel space of multilevel \((R,S_{\sigma })\)-commutative matrices

  • 期刊论文

  • 发表日期：

    2025年5月8日

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  • Ze-Kun Lyu,
  • Wei-Ru Xu

摘要

Multilevel matrix is a special kind of block matrices, each block has smaller blocks. In this paper, we discuss the necessary and sufficient conditions under which a matrix belongs to a kind of hierarchical structured matrices, and this kind of matrices commute with two c-involutory matrices. The singular value decomposition of such multilevel matrices are discussed. Moreover, we characterize the consistent solutions, which possess the structure described above, of the system \(AX=B, XC=D\) A X = B , X C = D under a kernel space constraint. A matrix nearness problem of these consistent solutions has been considered. A numerical algorithm and some numerical examples are presented to demonstrate the effectiveness and feasibility of our results.