Linear Algebra and its Applications最新文献

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On stabilizing index and cyclic index of certain amalgamated uniform hypergraphs 论某些汞齐均匀超图的稳定指数和循环指数

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-03 DOI: 10.1016/j.laa.2024.08.020

Cheng Yeaw Ku , Kok Bin Wong

Let G be a connected uniform hypergraph and $A (G)$ be the adjacency tensor of G. The largest absolute value of the eigenvalues of $A (G)$ is called the spectral radius of G. The number of eigenvectors of $A (G)$ associated with the spectral radius is called the stabilizing index of G. The number of eigenvalues of $A (G)$ with modulus equal to the spectral radius is called the cyclic index of G. In this paper, we consider a class of amalgamated uniform hypergraphs and compute its stabilizing index and cyclic index.

设 G 是一个连通的均匀超图，A(G) 是 G 的邻接张量。A(G) 的特征值的最大绝对值称为 G 的谱半径。模等于谱半径的 A(G) 特征值的个数称为 G 的循环指数。本文考虑一类汞齐均匀超图，并计算其稳定指数和循环指数。

引用次数: 0

The eigenvalue multiplicity of line graphs 线图的特征值多重性

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-03 DOI: 10.1016/j.laa.2024.08.021

Sarula Chang , Jianxi Li , Yirong Zheng

Let $m (G, λ)$ , $c (G)$ and $p (G)$ be the multiplicity of an eigenvalue λ, the cyclomatic number and the number of pendant vertices of a connected graph G, respectively. Yang et al. (2023) [10] proved that $m (L (T), λ) \leq p (T) - 1$ for any tree T, and characterized all trees T with $m (L (T), λ) = p (T) - 1$ , where $L (T)$ is the line graph of T. In this paper, we extend their result from a tree T to any graph $G \neq C_{n}$ , and prove that $m (L (G), λ) \leq 2 c (G) + p (G) - 1$ for any graph $G \neq C_{n}$ . Moreover, all graphs G with $m (L (G), - 1) = 2 c (G) + p (G) - 1$ are completely characterized.

设 m(G,λ)、c(G) 和 p(G) 分别为连通图 G 的特征值 λ 的倍率、循环数和挂顶点数。Yang 等人（2023）[10] 证明了对于任意树 T，m(L(T),λ)≤p(T)-1，并表征了 m(L(T),λ)=p(T)-1 的所有树 T，其中 L(T) 是 T 的线图。本文将他们的结果从树 T 扩展到任何图 G≠Cn，并证明对于任何图 G≠Cn，m(L(G),λ)≤2c(G)+p(G)-1。此外，m(L(G),-1)=2c(G)+p(G)-1 的所有图形 G 都是完全表征的。

引用次数: 0

Majorization in some symplectic weak supermajorizations 某些交点弱超大化中的大化

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-08-30 DOI: 10.1016/j.laa.2024.08.019

Shaowu Huang , Hemant K. Mishra

Symplectic eigenvalues are known to satisfy analogs of several classic eigenvalue inequalities. Of these is a set of weak supermajorization relations concerning symplectic eigenvalues that are weaker analogs of some majorization relations corresponding to eigenvalues. The aim of this letter is to establish necessary and sufficient conditions for the saturation of the symplectic weak supermajorization relations by majorization.

众所周知，交映特征值满足几个经典特征值不等式的类似条件。其中有一组关于交映特征值的弱超大化关系，它们是与特征值相对应的一些大化关系的弱类比。这封信的目的是为交点弱超大化关系的大化饱和建立必要和充分条件。

引用次数: 0

Normal approximations of commuting square-summable matrix families 可换算平方和矩阵族的正态近似值

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-08-30 DOI: 10.1016/j.laa.2024.08.017

Alexandru Chirvasitu

For any square-summable commuting family ${(A_{i})}_{i \in I}$ of complex $n \times n$ matrices there is a normal commuting family ${(B_{i})}_{i}$ no farther from it, in squared normalized $ℓ^{2}$ distance, than the diameter of the numerical range of $\sum_{i} A_{i}^{⁎} A_{i}$ . Specializing in one direction (limiting case of the inequality for finite I) this recovers a result of M. Fraas: if $\sum_{i = 1}^{ℓ} A_{i}^{⁎} A_{i}$ is a multiple of the identity for commuting $A_{i} \in M_{n} (C)$ then the $A_{i}$ are normal; specializing in another (singleton I) retrieves the well-known fact that close-to-isometric matrices are close to isometries.

对于任何复 n×n 矩阵的可平方和换向族 (Ai)i∈I，都有一个正态换向族 (Bi)i，其平方归一化 ℓ2 距离不远于 ∑iAi⁎Ai 数值范围的直径。Fraas 的一个结果：如果∑i=1ℓAi⁎Ai 是换元 Ai∈Mn(C)的等式的倍数，那么 Ai 是正交的；而从另一个方向（单子 I）来看，则可以得到一个众所周知的事实：接近等距矩阵接近于等距矩阵。

引用次数: 0

Contractivity of Möbius functions of operators 算子莫比乌斯函数的收缩性

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-08-29 DOI: 10.1016/j.laa.2024.08.018

Thomas Ransford , Dashdondog Tsedenbayar

Let T be an injective bounded linear operator on a complex Hilbert space. We characterize the complex numbers $λ, μ$ for which $(I + λ T) {(I + μ T)}^{- 1}$ is a contraction, the characterization being expressed in terms of the numerical range of the possibly unbounded operator $T^{- 1}$ .

When $T = V$ , the Volterra operator on $L^{2} [0, 1]$ , this leads to a result of Khadkhuu, Zemánek and the second author, characterizing those $λ, μ$ for which $(I + λ V) {(I + μ V)}^{- 1}$ is a contraction. Taking $T = V^{n}$ , we further deduce that $(I + λ V^{n}) {(I + μ V^{n})}^{- 1}$ is never a contraction if $n \geq 2$ and $λ \neq μ$ .

设 T 是复希尔伯特空间上的注入有界线性算子。当 T=V （L2[0,1] 上的 Volterra 算子）时，这将引出 Khadkhuu、Zemánek 和第二作者的一个结果，即描述那些 (I+λV)(I+μV)-1 是收缩的 λ,μ 的特征。以 T=Vn 为例，我们进一步推导出，如果 n≥2 且 λ≠μ 时，(I+λVn)(I+μVn)-1 绝不是收缩。

引用次数: 0

Matrix periods and competition periods of Boolean Toeplitz matrices II 布尔托普利兹矩阵的矩阵周期和竞争周期 II

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-08-28 DOI: 10.1016/j.laa.2024.08.016

Gi-Sang Cheon , Bumtle Kang , Suh-Ryung Kim , Homoon Ryu

<div>This paper is a follow-up to the paper of Cheon et al. (2023) [2]. Given subsets S and T of <math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>}</mo></math>, an <math><mi>n</mi><mo>×</mo><mi>n</mi></math> Toeplitz matrix <math><mi>A</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>〈</mo><mi>S</mi><mo>;</mo><mi>T</mi><mo>〉</mo></math> is defined to have 1 as the <math><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></math>-entry if and only if <math><mi>j</mi><mo>−</mo><mi>i</mi><mo>∈</mo><mi>S</mi></math> or <math><mi>i</mi><mo>−</mo><mi>j</mi><mo>∈</mo><mi>T</mi></math>. In the previous paper, we have shown that the matrix period and the competition period of Toeplitz matrices <math><mi>A</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>〈</mo><mi>S</mi><mo>;</mo><mi>T</mi><mo>〉</mo></math> satisfying the condition (⋆) <math><mi>max</mi><mo>⁡</mo><mi>S</mi><mo>+</mo><mi>min</mi><mo>⁡</mo><mi>T</mi><mo>≤</mo><mi>n</mi></math> and <math><mi>min</mi><mo>⁡</mo><mi>S</mi><mo>+</mo><mi>max</mi><mo>⁡</mo><mi>T</mi><mo>≤</mo><mi>n</mi></math> are <math><msup><mrow><mi>d</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>/</mo><mi>d</mi></math> and 1, respectively, where <math><msup><mrow><mi>d</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>s</mi><mo>+</mo><mi>t</mi><mo>|</mo><mi>s</mi><mo>∈</mo><mi>S</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>T</mi><mo>)</mo></math> and <math><mi>d</mi><mo>=</mo><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>d</mi><mo>,</mo><mi>min</mi><mo>⁡</mo><mi>S</mi><mo>)</mo></math>. In this paper, we claim that even if (⋆) is relaxed to the existence of elements <math><mi>s</mi><mo>∈</mo><mi>S</mi></math> and <math><mi>t</mi><mo>∈</mo><mi>T</mi></math> satisfying <math><mi>s</mi><mo>+</mo><mi>t</mi><mo>≤</mo><mi>n</mi></math> and <math><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn></math>, the same result holds. There are infinitely many Toeplitz matrices that do not satisfy (⋆) but the relaxed condition. For example, for any positive integers <math><mi>k</mi><mo>,</mo><mi>n</mi></math> with <math><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>≤</mo><mi>n</mi></math>, it is easy to see that <math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>〈</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>;</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>〉</mo></math> does not satisfy (⋆) but satisfies the relaxed condition. Furthermore, we show that the limit of the matrix sequence <math><msubsup><mrow><mo>{</mo><

本文是 Cheon 等人 (2023) [2] 论文的后续。给定{1,...,n-1}的子集 S 和 T，n×n 托普利兹矩阵 A=Tn〈S;T〉的定义是，当且仅当 j-i∈S 或 i-j∈T 时，（i,j）项为 1。在前一篇论文中，我们已经证明了满足条件（⋆）maxS+minT≤n 和 minS+maxT≤n 的托普利兹矩阵 A=Tn〈S;T〉的矩阵周期和竞争周期分别为 d+/d 和 1，其中 d+=gcd(s+t|s∈S,t∈T) 和 d=gcd(d,minS)。在本文中，我们声称，即使将 (⋆) 放宽到存在满足 s+t≤n 且 gcd(s,t)=1 的元素 s∈S 和 t∈T ，结果也同样成立。有无限多的托普利兹矩阵不满足 (⋆) 但满足放宽条件。例如，对于任何 2k+1≤n 的正整数 k、n，很容易看出 Tn〈k,n-k;k+1,n-k-1〉不满足 (⋆)，但满足松弛条件。此外，我们还证明矩阵序列 {Am(AT)m}m=1∞ 的极限是 Tn〈d+,2d+,...,⌊n/d+⌋d+〉。

引用次数: 0

Minimizing the Laplacian-energy-like of graphs 图的类拉普拉奇能量最小化

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-08-23 DOI: 10.1016/j.laa.2024.08.015

Gao-Xuan Luo, Shi-Cai Gong , Jing Tian

Let G be a connected simple graph with order n and Laplacian matrix $L (G)$ . The Laplacian-energy-like of G is defined as $L E L (G) = \sum_{i = 1}^{n} \sqrt{λ_{i}},$ where $λ_{i}$ is the eigenvalue of $L (G)$ for $i = 1, \dots, n$ . In this paper, with the aid of Ferrers diagrams of threshold graphs, we provide an algebraic combinatorial approach to determine the graphs with minimal Laplacian-energy-like among all connected graphs having n vertices and m edges, showing that the extremal graph is a special threshold graph, named as the quasi-complete graph.

设 G 是阶数为 n 的连通简单图，且有拉普拉斯矩阵 L(G)。G 的类拉普拉奇能量定义为：LEL(G)=∑i=1nλi，其中，λi 是 L(G) i=1，...，n 时的特征值。本文借助阈值图的费勒斯图，提供了一种代数组合方法，以确定在具有 n 个顶点和 m 条边的所有连通图中具有最小拉普拉奇能样的图，证明极值图是一种特殊的阈值图，命名为准完全图。

引用次数: 0

Unknotting nonorientable surfaces of genus 4 and 5 解结属 4 和 5 的非定向曲面

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-08-22 DOI: 10.1016/j.laa.2024.08.014

Mark Pencovitch

Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in $D^{4}$ with knot group $Z_{2}$ .

In particular we show that if two such surfaces have the same normal Euler number, the same fixed knot boundary K such that $| \det (K) | = 1$ , and the same nonorientable genus 4 or 5, then they are ambiently isotopic rel. boundary.

This implies that closed, nonorientable, locally flatly embedded surfaces in the 4-sphere with knot group $Z_{2}$ of nonorientable genus 4 and 5 are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that an obstruction to modified surgery is elementary. Furthermore we show that this method fails for nonorientable genus 6 and 7.

在康威、奥森和鲍威尔工作的基础上，我们研究了结群为 Z2 的 D4 中不可定向、紧凑、局部平嵌曲面的同位类相对边界。特别是，我们证明了如果两个这样的曲面具有相同的法欧拉数、相同的固定结边界 K（使得 |det(K)|=1 ）以及相同的不可定向属 4 或 5，那么它们就是环境同位类相对边界。这意味着在 4 球中，具有不可定向的结群 Z2 的不可定向属 4 和 5 的封闭、不可定向、局部平嵌曲面在拓扑上是无结的。证明依赖于在 Sage 中实现的计算，这意味着修正手术的障碍是基本的。此外，我们还证明了这种方法对于不可定向的属 6 和属 7 不适用。

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引用次数: 0

Even grade generic skew-symmetric matrix polynomials with bounded rank 有界秩的偶级通用偏斜对称矩阵多项式

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-08-22 DOI: 10.1016/j.laa.2024.07.024

Fernando De Terán , Andrii Dmytryshyn , Froilán M. Dopico

We show that the set of $m \times m$ complex skew-symmetric matrix polynomials of even grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic $m \times m$ complex skew-symmetric matrix polynomials of even grade d and rank at most 2r. The analogous problem for the case of skew-symmetric matrix polynomials of odd grade is solved in [24].

我们证明，偶数级 d 的 m×m 复数偏斜对称矩阵多项式集合，即最多 d 级和（正常）最多 2r 级，是具有某些明确描述的完整特征结构的单个矩阵多项式集合的闭集。这个完整的特征结构对应于偶数级 d、秩至多 2r 的最一般的 m×m 复数偏斜对称矩阵多项式。奇数级的倾斜对称矩阵多项式的类似问题已在 [24] 中解决。

引用次数: 0

New scattered linearized quadrinomials 新的分散线性化四则运算

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-08-21 DOI: 10.1016/j.laa.2024.08.012

Valentino Smaldore , Corrado Zanella , Ferdinando Zullo

<div>Let <math><mn>1</mn><mo><</mo><mi>t</mi><mo><</mo><mi>n</mi></math> be integers, where t is a divisor of n. An <math><mi>R-</mi><mspace></mspace><msup><mrow><mi>q</mi></mrow><mrow><mi>t</mi></mrow></msup></math>-partially scattered polynomial is a q-polynomial f in <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math> that satisfies the condition that for all <math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math> such that <math><mi>x</mi><mo>/</mo><mi>y</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></msub></math>, if <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>/</mo><mi>x</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>/</mo><mi>y</mi></math>, then <math><mi>x</mi><mo>/</mo><mi>y</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>; f is called scattered if this implication holds for all <math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math>. Two polynomials in <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math> are said to be equivalent if their graphs are in the same orbit under the action of the group <math><mrow><mi>Γ</mi><mi>L</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math>. For <math><mi>n</mi><mo>></mo><mn>8</mn></math> only three families of scattered polynomials in <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math> are known: (i) monomials of pseudoregulus type, <math><mo>(</mo><mi>i</mi><mi>i</mi><mo>)</mo></math> binomials of Lunardon-Polverino type, and <math><mo>(</mo><mi>i</mi><mi>i</mi><mi>i</mi><mo>)</mo></math> a family of quadrinomials defined in [1], [10] and extended in [8], [13]. In this paper we prove that the polynomial <math><msub><mrow><mi>φ</mi></mrow><mrow><mi>m</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>J</mi></mrow></msup></mrow></msub><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>J</mi><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup></mrow></msup><mo>+</mo><

设 1<t<n 为整数，其中 t 是 n 的除数。一个 R-qt 部分分散多项式是 Fqn[X] 中的一个 q 多项式 f，它满足以下条件：对于所有 x，y∈Fqn⁎，使得 x/y∈Fqt，如果 f(x)/x=f(y)/y，那么 x/y∈Fq；如果这个蕴涵对于所有 x，y∈Fqn⁎都成立，那么 f 称为分散多项式。如果在群ΓL(2,qn)的作用下，Fqn[X] 中的两个多项式的图在同一轨道上，则称这两个多项式等价。对于 n>8，已知 Fqn[X] 中只有三个分散多项式族：(i) pseudoregulus 型单项式，(ii) Lunardon-Polverino 型二项式，以及 (iii) 在 [1], [10] 中定义并在 [8], [13] 中扩展的四次多项式族。在本文中，我们证明多项式 φm,qJ=XqJ(t-1)+XqJ(2t-1)+m(XqJ-XqJ(t+1))∈Fq2t[X], q 奇，t≥3 对于 m∈Fqt⁎ 和 J 与 2t 共素数的每一个值都是 R-qt 部分分散的。此外，对于每一个 t>4 和 q>5，都存在这样的 m 值，即φm,q 对于上述 (i)、(ii) 和 (iii) 中提到的多项式是分散的和新的。相关的线性集合至少有两个 ΓL 类。

引用次数: 0

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数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Linear Algebra and its Applications

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