Linear Algebra and its Applications最新文献

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On Kemeny's constant and stochastic complement 关于凯梅尼常数和随机补数

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-05 DOI: 10.1016/j.laa.2024.09.001

Given a stochastic matrix P partitioned in four blocks $P_{i j}$ , $i, j = 1, 2$ , Kemeny's constant $κ (P)$ is expressed in terms of Kemeny's constants of the stochastic complements $P_{1} = P_{11} + P_{12} {(I - P_{22})}^{- 1} P_{21}$ , and $P_{2} = P_{22} + P_{21} {(I - P_{11})}^{- 1} P_{12}$ . Specific cases concerning periodic Markov chains and Kronecker products of stochastic matrices are investigated. Bounds to Kemeny's constant of perturbed matrices are given. Relying on these theoretical results, a divide-and-conquer algorithm for the efficient computation of Kemeny's constant of graphs is designed. Numerical experiments performed on real world problems show the high efficiency and reliability of this algorithm.

鉴于随机矩阵 P 分成四个块 Pij（i,j=1,2），可门尼常数 κ(P)用随机补集 P1=P11+P12(I-P22)-1P21 和 P2=P22+P21(I-P11)-1P12 的可门尼常数表示。研究了周期马尔可夫链和随机矩阵的克朗克积的具体情况。给出了扰动矩阵的凯美尼常数的界值。根据这些理论结果，设计了一种高效计算图的凯门尼常数的分而治之算法。在实际问题上进行的数值实验表明了该算法的高效性和可靠性。

引用次数: 0

Some integer values in the spectra of burnt pancake graphs 烧饼图谱中的一些整数值

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-05 DOI: 10.1016/j.laa.2024.09.003

The burnt pancake graph, denoted by ${BP}_{n}$ , is formed by connecting signed permutations via prefix reversals. Here, we discuss some spectral properties of ${BP}_{n}$ . More precisely, we prove that the adjacency spectrum of ${BP}_{n}$ contains all integer values in the set ${0, 1, \dots, n} ∖ {⌊ n / 2 ⌋}$ .

烧饼图（用 BPn 表示）是通过前缀反转连接有符号的排列而形成的。在此，我们将讨论 BPn 的一些谱属性。更确切地说，我们证明 BPn 的邻接谱包含集合 {0,1,...n}∖{⌊n/2⌋} 中的所有整数值。

引用次数: 0

The n-th production matrix of a Riordan array 瑞尔丹数组的 n 次生产矩阵

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-09-04 DOI: 10.1016/j.laa.2024.08.022

The production matrix plays an important role in characterizing a Riordan array. Recently, Barry explored the notion of the n-th production matrix and characterized the Riordan arrays corresponding to the second and third production matrices respectively. This paper is devoted to study the n-th production matrix and its corresponding Riordan arrays systematically. Our work is threefold. First, we show that every n-th production matrix can be factorized into a product of n matrices associated with the ordinary production matrix. Second, we prove a characterization of the Riordan array corresponding to the n-th production matrix, which was conjectured by Barry. Third, we claim that if the ordinary production matrix of a Riordan array is totally positive, so are the n-th production matrix and its corresponding Riordan arrays. Our results are illustrated by the generalized Catalan array which includes many well-known Riordan arrays as special cases.

生成矩阵在描述瑞奥德恩数组的特征时起着重要作用。最近，巴里探索了 nth 生产矩阵的概念，并分别描述了与第二和第三生产矩阵相对应的瑞尔丹数组。本文致力于系统研究 nth 生产矩阵及其对应的瑞尔丹数组。我们的工作包括三个方面。首先，我们证明了每个 n 次生产矩阵都可以因式分解为与普通生产矩阵相关的 n 个矩阵的乘积。其次，我们证明了与第 n 个生产矩阵相对应的瑞尔丹数组的特征，这是巴里的猜想。第三，我们声称，如果一个瑞尔丹数组的普通生产矩阵是全正的，那么第 n 个生产矩阵及其对应的瑞尔丹数组也是全正的。我们的结果通过广义加泰罗尼亚数组来说明，其中包括许多著名的瑞尔丹数组特例。

引用次数: 0

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

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

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

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

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

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

<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

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

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