The paper introduces and considers two new classes of the so-called S-SDD<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$_k(\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>k</mi> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and S-GSDD<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$_k(\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>k</mi> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> matrices, where S is a nonempty subset of the set <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$R_A$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>A</mi> </msub> </math></EquationSource> </InlineEquation> of strictly diagonally dominant rows of A, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$k\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\sigma \in (0,\;1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.277778em" /> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> is a real parameter. Properties of these matrices and their interrelations with some other matrix classes are considered. In particular, it is shown that S-SDD<InlineEquation ID="IEq10"> <EquationSource Format="TEX">$_k(\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>k</mi> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and S-GSDD<InlineEquation ID="IEq11"> <EquationSource Format="TEX">$_k(\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>k</mi> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> matrices are nonsingular <InlineEquation ID="IEq12"> <EquationSource Format="TEX">${\mathcal {H}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>-matrices and, moreover, SD-SDD and S-SSDD (Schur SDD) matrices. Based on the latter result, a general parameter-free upper bound for the <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$l_\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>-norms of the inverses to S-SDD<InlineEquation ID="IEq14"> <EquationSource Format="TEX">$_k(\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>k</mi> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and S-GSDD<InlineEquation ID="IEq15"> <EquationSource Format="TEX">$_k(\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>k</mi> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> matrices is obtained. Also upper bounds for <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\Vert A^{-1}\Vert _\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msup> <mi>A</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mi>∞</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> based on a specific diagonal scaling of S-SDD<InlineEquation ID="IEq17"> <EquationSource Format="TEX">$_k(\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>k</mi> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and S-SDD<InlineEquation ID="IEq18"> <EquationSource Format="TEX">$_k(\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>k</mi> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> matrices A, directly related to their definitions, are provided. Bibliography: 14 titles.

ON S-SDD$_k(\sigma )$ AND S-GSDD$_k(\sigma )$ MATRICES

期刊论文
发表日期：

2026年5月19日

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L. Yu. Kolotilina

摘要

The paper introduces and considers two new classes of the so-called S-SDD $_k(\sigma )$ $_{k} (σ)$ and S-GSDD $_k(\sigma )$ $_{k} (σ)$ matrices, where S is a nonempty subset of the set $R_A$ $R_{A}$ of strictly diagonally dominant rows of A, $k\ge 1$ $k \geq 1$ , and $\sigma \in (0,\;1]$ $σ \in (0, 1]$ is a real parameter. Properties of these matrices and their interrelations with some other matrix classes are considered. In particular, it is shown that S-SDD $_k(\sigma )$ $_{k} (σ)$ and S-GSDD $_k(\sigma )$ $_{k} (σ)$ matrices are nonsingular ${\mathcal {H}}$ $H$ -matrices and, moreover, SD-SDD and S-SSDD (Schur SDD) matrices. Based on the latter result, a general parameter-free upper bound for the $l_\infty $ $l_{\infty}$ -norms of the inverses to S-SDD $_k(\sigma )$ $_{k} (σ)$ and S-GSDD $_k(\sigma )$ $_{k} (σ)$ matrices is obtained. Also upper bounds for $\Vert A^{-1}\Vert _\infty $ $‖ A^{- 1} ‖_{\infty}$ based on a specific diagonal scaling of S-SDD $_k(\sigma )$ $_{k} (σ)$ and S-SDD $_k(\sigma )$ $_{k} (σ)$ matrices A, directly related to their definitions, are provided. Bibliography: 14 titles.

ON S-SDD\(_k(\sigma )\) AND S-GSDD\(_k(\sigma )\) MATRICES

摘要