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<p>The weighted subdirect sum of matrices plays a fundamental role in applications involving overlapping structures, such as multilayer network analysis and overlapping block iterative methods for solving linear systems. In this paper, some sufficient conditions are provided to ensure that the weighted subdirect sum of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(GSDD_1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <mi>D</mi> <msub> <mi>D</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> matrices is in the class of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(GSDD_1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <mi>D</mi> <msub> <mi>D</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> matrices. We present, in particular, that the weighted 1-subdirect sum of a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(GSDD_1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <mi>D</mi> <msub> <mi>D</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> matrix and a strictly diagonally dominant matrix remains a <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(GSDD_1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <mi>D</mi> <msub> <mi>D</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> matrix. Theoretical results are complemented by numerical examples demonstrating the validity and applicability of the proposed conditions.</p>

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Weighted subdirect sum of \(GSD{D_1}\) matrices

  • 期刊论文

  • 发表日期：

    2026年7月11日

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  • Xi Wu,
  • Deshu Sun,
  • Yingxia Zhao

摘要

The weighted subdirect sum of matrices plays a fundamental role in applications involving overlapping structures, such as multilayer network analysis and overlapping block iterative methods for solving linear systems. In this paper, some sufficient conditions are provided to ensure that the weighted subdirect sum of \(GSDD_1\) G S D D 1 matrices is in the class of \(GSDD_1\) G S D D 1 matrices. We present, in particular, that the weighted 1-subdirect sum of a \(GSDD_1\) G S D D 1 matrix and a strictly diagonally dominant matrix remains a \(GSDD_1\) G S D D 1 matrix. Theoretical results are complemented by numerical examples demonstrating the validity and applicability of the proposed conditions.