Linear Algebra and its Applications最新文献

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Convergence analysis of optimal SOR for a class of consistently ordered 2-cyclic matrices with complex spectra 一类复谱一致有序2循环矩阵最优SOR的收敛性分析

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-10-21 DOI: 10.1016/j.laa.2025.10.019

L. Robert Hocking, Chen Greif

Asymptotic rates of convergence of optimal SOR applied to linear systems with consistently ordered 2-cyclic matrices have been extensively studied in the case where the Jacobi spectrum is real and contained in an interval centered at the origin. It is well known that as the rightmost endpoint of the interval approaches 1 from below, optimal SOR converges an order of magnitude faster than Jacobi. We generalize this to the situation where the Jacobi spectrum is contained in a line segment in the complex plane that is symmetric about the origin. This is an important class of linear systems, which arise often in various physical applications; complex-shifted linear systems are included in this family. Optimal relaxation parameters are known in this case, but a detailed convergence analysis does not seem to exist in the literature. Using techniques of complex analysis, we derive convergence rates, finding that in the complex case they are affected not only by the distance to 1 of the right-hand endpoint of the line segment as in the real case, but also by its phase. We then exploit a useful convergence property that our analysis reveals to generate a hybrid SOR–multigrid scheme and demonstrate its merits as a shifted Laplacian preconditioner for GMRES applied to the Helmholtz equation.

在雅可比谱为实数且包含在以原点为中心的区间内的情况下，广泛研究了最优SOR的渐近收敛速率，并将其应用于具有一致序2循环矩阵的线性系统。众所周知，当区间的最右端点从下接近1时，最优SOR的收敛速度比Jacobi快一个数量级。我们把它推广到雅可比谱包含在复平面的线段中关于原点对称的情况。这是一类重要的线性系统，经常出现在各种物理应用中；复移线性系统也属于这一类。在这种情况下，已知最优松弛参数，但文献中似乎没有详细的收敛分析。利用复变分析的技术，我们推导出收敛速率，发现在复变情况下，收敛速率不仅受到实际情况中线段右端点到1的距离的影响，而且还受到其相位的影响。然后，我们利用我们的分析揭示的有用的收敛性质来生成混合sor -多网格方案，并证明其作为GMRES应用于亥姆霍兹方程的移位拉普拉斯预条件的优点。

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引用次数: 0

Involutions and angles between subspaces 子空间之间的对合和夹角

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-10-20 DOI: 10.1016/j.laa.2025.10.017

Jean-Christophe Bourin , Eun-Young Lee

If S is an involution matrix, which means

S^{2} = I

, then S is unitarily equivalent to

(ε I_{k}) \oplus {⨁_{i = 1}^{m} (\begin{matrix} 0 & x_{i}^{- 1} \\ x_{i} & 0 \end{matrix})}

where

ε = \pm 1

I_{k}

is the identity of size

k = | Tr S |

, and

0 < x_{i} \leq 1

i = 1, \dots, m = n - k

. This can be used to introduce principal angles between subspaces and yields formulas for the angles between the eigenspaces of S such as

\sin α_{i} = 2 / (x_{i} + x_{i}^{- 1})

如果S是一个对合矩阵，即S2=I，则S酉等价于(εIk)⊕{λ I =1m(0xi - 1xi0)}，其中ε=±1，Ik是大小k=|TrS|的恒等式，且0<；xi≤1,I =1，…，m=n - k。这可以用来引入子空间之间的主角，并给出S的特征空间之间的角的公式，如sin (αi) =2/(xi+xi - 1)。

引用次数: 0

Gershgorin type inclusion-exclusion sets for matrix polynomials 矩阵多项式的Gershgorin型包容-排斥集

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-10-20 DOI: 10.1016/j.laa.2025.10.012

Christina Michailidou, Vasiliki Panagakou, Panayiotis Psarrakos

In this paper, improvements of the spectrum estimation for matrix polynomials given by the Gershgorin set, the Brauer set, and the Dashnic-Zusmanovich set are derived by substracting regions of the complex plane which do not contain eigenvalues. Geometrical and topological properties of the exclusion sets are obtained, and illustrative examples are presented.

本文通过减去复平面上不包含特征值的区域，得到了Gershgorin集、Brauer集和Dashnic-Zusmanovich集给出的矩阵多项式谱估计的改进。得到了不相容集的几何和拓扑性质，并给出了实例。

引用次数: 0

Clique complexes of strongly regular graphs, their eigenvalues, and cohomology groups 强正则图的团复，它们的特征值和上同调群

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-10-20 DOI: 10.1016/j.laa.2025.10.013

Sebastian M. Cioabă , Krystal Guo , Chunxu Ji , Mutasim Mim

It is known that non-isomorphic strongly regular graphs with the same parameters must be cospectral (have the same eigenvalues). In this paper, we investigate whether the spectra of higher order Laplacians associated with these graphs can distinguish them. In this direction, we study the clique complexes of strongly regular graphs, and determine the spectra of the triangle complexes of several families of strongly regular graphs including Hamming graphs and Triangular graphs. In many cases, the spectrum of the triangle complex distinguishes between strongly regular graphs with the same parameters, but we find some examples where that is not the case. We also prove that if a graph has the property that for any induced cycle, there are four consecutive vertices on the cycle with a common neighbor, then the first cohomology group of the graph is trivial and we apply this result to several families of graphs.

已知具有相同参数的非同构强正则图必须是共谱的（具有相同的特征值）。在本文中，我们研究了与这些图相关的高阶拉普拉斯谱是否可以区分它们。在此方向上，我们研究了强正则图的团复形，并确定了包括Hamming图和三角图在内的几类强正则图的三角复形的谱。在许多情况下，三角复形的谱区分具有相同参数的强正则图，但我们发现一些例子并非如此。我们还证明了如果一个图具有这样的性质：对于任何诱导环，在这个环上有4个连续的顶点有一个共同的邻居，那么这个图的第一个上同群是平凡的，我们将这个结果应用于几个图族。

引用次数: 0

Uniform hypertrees with maximum nullity 具有最大零的一致超树

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-10-17 DOI: 10.1016/j.laa.2025.10.016

Ya-Nan Zheng

Let

A

be the adjacency tensor of a k-uniform hypergraph H. The nullity of H is the multiplicity of the eigenvalue zero in the spectrum of H, i.e., the algebraic multiplicity of the eigenvalue zero of

A

. A connected and acyclic hypergraph is called a hypertree. In this paper, by exploring the relationship between the nullity of k-uniform hypertrees and the nullity of their subhypergraphs, we study the extremal nullity of k-uniform hypertrees. We prove that the k-uniform hyperstar

S_{m}^{k}

attains uniquely the maximum nullity among all k-uniform hypertrees with m edges.

设A为k-一致超图H的邻接张量，H的零性是H谱中特征值零的多重性，即A的特征值零的代数多重性。连通无环超图称为超树。本文通过探讨k-一致超树的零性与其子超图的零性之间的关系，研究了k-一致超树的极值零性。证明了k-一致超星Smk在所有有m条边的k-一致超树中唯一地达到极大零。

引用次数: 0

Reflections in L2(T) L2(T)中的反射

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-10-17 DOI: 10.1016/j.laa.2025.10.014

Esteban Andruchow

<div><div>Let <math><mi>D</mi><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo><mi>C</mi><mo>:</mo><mo>|</mo><mi>z</mi><mo>|</mo><mo><</mo><mn>1</mn><mo>}</mo></math> and <math><mi>T</mi><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo><mi>C</mi><mo>:</mo><mo>|</mo><mi>z</mi><mo>|</mo><mo>=</mo><mn>1</mn><mo>}</mo></math>. For <math><mi>a</mi><mo>∈</mo><mi>D</mi></math>, consider <math><msub><mrow><mi>φ</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mo>−</mo><mi>z</mi></mrow><mrow><mn>1</mn><mo>−</mo><mover><mrow><mi>a</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>z</mi></mrow></mfrac></math> and <math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub></math> the composition operator in <math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>T</mi><mo>)</mo></math> induced by <math><msub><mrow><mi>φ</mi></mrow><mrow><mi>a</mi></mrow></msub></math>:<math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub><mi>f</mi><mo>=</mo><mi>f</mi><mo>∘</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>.</mo></math> Clearly <math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub></math> satisfies <math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>=</mo><mi>I</mi></math>, i.e., is a non-selfadjoint reflection. We also consider the following symmetries (selfadjoint reflections) related to <math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub></math>:<math><msub><mrow><mi>R</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mfrac><mrow><mo>|</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>|</mo></mrow><mrow><msub><mrow><mo>‖</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>‖</mo></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub><mspace></mspace><mtext> and </mtext><mspace></mspace><msub><mrow><mi>W</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msub></mrow><mrow><msub><mrow><mo>‖</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>‖</mo></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>,</mo></math> where <math><msub><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>−</mo><mover><mrow><mi>a</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>z</mi></mrow></mfrac></math> is the Szego kernel. The symmetry <math><msub><mrow><mi>R</mi></mrow><mrow><mi>a</

设D={z∈C:|z|<1}， T={z∈C:|z|=1}。对于a∈D，考虑φa(z)=a−z1−a¯z和Ca是由φa引起的L2(T)中的复合算子：Caf=f°φa。显然，Ca满足Ca2=I，即是一个非自伴反射。我们还考虑了以下与Ca相关的对称性（自伴随反射）：Ra=M|ka|‖ka‖2Ca和Wa=Mka‖ka‖2Ca，其中ka(z)=11−a¯z是Szego核。对称性Ra是Ca的极分解中的酉部分。我们对Ta=Ca、Ra或Wa的特征空间N（Ta±I）进行了表征，并研究了它们在改变参数a时的相对位置，例如，对于a≠b∈D， N(Ta±I)∩N(Tb±I)⊥，N(Ta±I)⊥∩N(Tb±I)⊥，N(Ta±I)⊥∩N(Tb±I)⊥，等等。

引用次数: 0

Weighted parallelogram law, polarization identity, and Clarkson-McCarthy inequalities 加权平行四边形定律，极化同一性和克拉克森-麦卡锡不等式

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-10-16 DOI: 10.1016/j.laa.2025.10.015

Fuad Kittaneh , Jorge Losada , Bashar Mayyas

The aim of this article is to give weighted versions of the Clarkson-McCarthy inequalities for matrices. In addition, we present weighted versions of the parallelogram law and the polarization identity for matrices.

本文的目的是给出矩阵的克拉克森-麦卡锡不等式的加权版本。此外，我们给出了平行四边形定律的加权版本和矩阵的极化恒等式。

引用次数: 0

Convexity of sums of eigenvalues of a segment of unitaries 一元段的特征值和的凸性

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-10-15 DOI: 10.1016/j.laa.2025.10.011

Gabriel Larotonda , Martin Miglioli

For an

n \times n

unitary matrix

u = e^{z}

with z skew-Hermitian, the angles of u are the arguments of its spectrum, i.e. the spectrum of

- i z

. For

1 \leq m \leq n

, we show that

s_{m} (t)

, the sum of the first m angles of the path

t \mapsto e^{t x} e^{y}

of unitary matrices, is a convex function of t (provided the path stays in a vicinity of the identity matrix). This vicinity is described in terms of the operator norm of matrices, and it is optimal. We show that when all the maps

t \mapsto s_{m} (t)

are linear, then x commutes with y. Several applications to unitarily invariant norms in the unitary group are given. Then we extend these applications to Ad-invariant Finsler norms in the special unitary group of matrices. This last result is obtained by proving that any Ad-invariant Finsler norm in a compact semi-simple Lie group K is the supremum of a family of what we call orbit norms, induced by the Killing form of K.

对于具有z斜厄米矩阵的n×n酉矩阵u=ez， u的角是其谱的参数，即- iz的谱。对于1≤m≤n，我们证明了sm(t)，即幺正矩阵的路径t的前m个角的和，是t的凸函数（假设路径在单位矩阵附近）。这个邻近用矩阵的算子范数来描述，它是最优的。我们证明了当所有映射t∈sm(t)都是线性时，则x可以与y交换。给出了在酉群中酉不变范数的几个应用。然后将这些应用推广到特殊酉矩阵群中的常不变Finsler范数。最后一个结果是通过证明紧半单李群K中的任意不变Finsler范数是由K的Killing形式导出的一组轨道范数的上零点而得到的。

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引用次数: 0

Multiple-term refinements and reverses of real power form for Young-type inequalities via weak submajorization type theorems 基于弱次多数型定理的young型不等式实权形式的多项改进和反演

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-10-10 DOI: 10.1016/j.laa.2025.10.004

Duong Quoc Huy , Thai Thuan Quang , Doan Thi Thuy Van

In a recent paper, D. Q. Huy, D. T. T. Van, D. T. Xinh (2023) [11] utilized the theory of weak submajorization to propose refinements and reverses of real power form for Young-type inequalities which are much better than previous results. This method is really effective when the number of terms in the considered inequalities is not too much, but it seems to be useless in general case. In this paper, we develop weak submajorization type theorems to offer multiple-term refinements and reverses of real power form for Young-type inequalities. These new developments also give us a fresh look at the relationship between the classical theory of weak submajorization and majorization. In addition, complete refinements and reverses of real power form for operator and matrix Young-type inequalities have also been established.

在最近的一篇论文中，D. Q. Huy， D. T. Van， D. T. Xinh(2023)[11]利用弱次多数理论提出了Young-type不等式的实权形式的改进和反转，比以前的结果好得多。当所考虑的不等式中的项数不太多时，这种方法非常有效，但在一般情况下似乎是无用的。在本文中，我们发展了弱次多数型定理，为young型不等式提供了实权形式的多项改进和反转。这些新进展也使我们对经典的弱次多数化理论和多数化理论之间的关系有了新的认识。此外，还建立了算子和矩阵young型不等式实幂形式的完全细化和反转。

引用次数: 0

Numerical instability of algebraic rootfinders 代数寻根器的数值不稳定性

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-10-10 DOI: 10.1016/j.laa.2025.10.009

Emil Graf, Alex Townsend

We demonstrate that the most popular variants of all common algebraic multidimensional rootfinding algorithms are unstable by analyzing the conditioning of subproblems that are constructed at intermediate steps. In particular, we give multidimensional polynomial systems for which the conditioning of a subproblem can be worse than the conditioning of the original problem by a factor that grows exponentially with the number of variables.

通过分析在中间步骤构造的子问题的条件，我们证明了所有常见的代数多维寻根算法的最流行变体是不稳定的。特别地，我们给出了多维多项式系统，其中子问题的条件条件可能比原问题的条件条件差，其因素随变量数量呈指数增长。

引用次数: 0

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Linear Algebra and its Applications

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