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<p>The paper introduces into consideration a new matrix class of the so-called SSDD (Schur SDD) matrices, which contains the class of SDD (strictly diagonally dominant) matrices and is itself contained in the class of nonsingular ℋ-matrices. The definition of an SSDD matrix <i>A</i> is based on distinguishing a subset <i>S</i> of its strictly diagonally dominant rows and requiring that the Schur complement <i>ℳ</i>(<i>A</i>)/<i>S</i> of its comparison matrix be an SDD matrix. Properties of SSDD matrices and their relations with other subclasses of the class of ℋ-matrices are considered. In particular, it is shown that such known matrix classes as those of OB, SOB, DZ, DZT (DZ-type), CKV-type, <i>S</i>-SDD, SDD<sub>1</sub>, SDD<sub><i>k</i></sub>, GSDD<sub>1</sub>, and also <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7711_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\({\text{GSDD}}_{1}^{*}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mtext>GSDD</mtext> <mrow> <mn>1</mn> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation> matrices all are contained in the class of SSDD matrices. On the other hand, the SSDD matrices themselves are simultaneously PH- and SDSDD matrices and, up to symmetric row and column permutations, they coincide with the block 2 × 2 generalized Nekrasov matrices, the so-called GN matrices. Also some upper bounds for the <i>l</i><sub>∞</sub>-norm of the inverse to an SSDD matrix are established.</p>

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SSDD Matrices and Relations with Other Subclasses of the Nonsingular ℋ-Matrices

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • L. Yu. Kolotilina

摘要

The paper introduces into consideration a new matrix class of the so-called SSDD (Schur SDD) matrices, which contains the class of SDD (strictly diagonally dominant) matrices and is itself contained in the class of nonsingular ℋ-matrices. The definition of an SSDD matrix A is based on distinguishing a subset S of its strictly diagonally dominant rows and requiring that the Schur complement ℳ(A)/S of its comparison matrix be an SDD matrix. Properties of SSDD matrices and their relations with other subclasses of the class of ℋ-matrices are considered. In particular, it is shown that such known matrix classes as those of OB, SOB, DZ, DZT (DZ-type), CKV-type, S-SDD, SDD1, SDDk, GSDD1, and also \({\text{GSDD}}_{1}^{*}\) GSDD 1 ∗ matrices all are contained in the class of SSDD matrices. On the other hand, the SSDD matrices themselves are simultaneously PH- and SDSDD matrices and, up to symmetric row and column permutations, they coincide with the block 2 × 2 generalized Nekrasov matrices, the so-called GN matrices. Also some upper bounds for the l∞-norm of the inverse to an SSDD matrix are established.