Linear Algebra and its Applications最新文献

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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

On the rank structure of the Moore-Penrose inverse of singular k-banded matrices 关于奇异 k 带状矩阵的摩尔-彭罗斯逆的秩结构

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-08-21 DOI: 10.1016/j.laa.2024.08.011

M.I. Bueno , Susana Furtado

It is well-established that, for an $n \times n$ singular k-banded complex matrix B, the submatrices of the Moore-Penrose inverse $B^{†}$ of B located strictly below (resp. above) its kth superdiagonal (resp. kth subdiagonal) have a certain bounded rank s depending on n, k and rankB. In this case, $B^{†}$ is said to satisfy a semiseparability condition. In this paper our focus is on singular strictly k-banded complex matrices B, and we show that the Moore-Penrose inverse of such a matrix satisfies a stronger condition, called generator representability. This means that there exist two matrices of rank at most s whose parts strictly below the kth diagonal (resp. above the kth subdiagonal) coincide with the same parts of $B^{†}$ . When $n \geq 3 k$ , we prove that s is precisely the minimum rank of these two matrices. We also illustrate through examples that when $n < 3 k$ those matrices may have rank less than s.

对于 n×n 奇异 k 带复矩阵 B，B 的摩尔-彭罗斯逆 B† 的子矩阵严格位于其第 k 个超对角线（或第 k 个次对角线）的下方（或上方），具有一定的有界秩 s，该秩取决于 n、k 和 rankB。在这种情况下，B† 满足半可分性条件。本文的重点是奇异的严格 k 带复矩阵 B，我们将证明这种矩阵的摩尔-彭罗斯逆满足一个更强的条件，即生成器可表示性。这意味着存在两个秩最多为 s 的矩阵，它们的第 k 条对角线以下（或第 k 条对角线以上）部分与 B† 的相同部分重合。当 n≥3k 时，我们证明 s 正是这两个矩阵的最小秩。我们还通过实例说明，当 n<3k 时，这些矩阵的秩可能小于 s。

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引用次数: 0

The determinant of {±1}-matrices and oriented hypergraphs {±1}矩阵和定向超图的行列式

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-08-20 DOI: 10.1016/j.laa.2024.08.013

Lucas J. Rusnak , Josephine Reynes , Russell Li , Eric Yan , Justin Yu

The determinants of ${\pm 1}$ -matrices are calculated via the oriented hypergraphic Laplacian and summing over incidence generalizations of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of 0 to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family.

{±1}矩阵的行列式是通过定向超图拉普拉卡和顶点循环覆盖的入射概括求和计算得出的。这些循环覆盖被签名，并根据它们的超边包含性划分为族。结果表明，每个非边缘单值族对拉普拉卡方的净贡献值为 0，而每个边缘单值族的总和为原始入射矩阵行列式的绝对值。我们还确定了简单的对称性及其与哈达玛最大行列式问题的关系。最后，只需使用边缘单子族循环覆盖的邻接最小集的符号，就能重新获得入射矩阵的条目。

引用次数: 0

Complete equitable decompositions 完整的公平分解

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-08-14 DOI: 10.1016/j.laa.2024.08.008

Joseph Drapeau , Joseph Henderson , Peter Seely , Dallas Smith , Benjamin Webb

A classical result in spectral graph theory states that if a graph G has an equitable partition π then the eigenvalues of the divisor graph $G_{π}$ are a subset of its eigenvalues, i.e. $σ (G_{π}) \subseteq σ (G)$ . A natural question is whether it is possible to recover the remaining eigenvalues $σ (G) - σ (G_{π})$ in a similar manner. Here we show that any weighted undirected graph with nontrivial equitable partition can be decomposed into a number of subgraphs whose collective spectra contain these remaining eigenvalues. Using this decomposition, which we refer to as a complete equitable decomposition, we introduce an algorithm for finding the eigenvalues of an undirected graph (symmetric matrix) with a nontrivial equitable partition. Under mild assumptions on this equitable partition we show that we can find eigenvalues of such a graph faster using this method when compared to standard methods. This is potentially useful as many real-world data sets are quite large and have a nontrivial equitable partition.

谱图理论中的一个经典结果表明，如果一个图 G 有一个公平分割 π，那么除法图 Gπ 的特征值就是其特征值的一个子集，即 σ(Gπ)⊆σ(G)。一个自然的问题是，是否有可能以类似的方式恢复其余的特征值 σ(G)-σ(Gπ)。在这里，我们将证明，任何具有非三等分的加权无向图都可以分解成若干子图，这些子图的集合谱包含这些剩余特征值。利用这种分解（我们称之为完全公平分解），我们引入了一种算法，用于找到具有非难公平分区的无向图（对称矩阵）的特征值。在对这种公平分区的温和假设下，我们证明，与标准方法相比，使用这种方法可以更快地找到这种图的特征值。这一点非常有用，因为现实世界中的许多数据集都相当大，而且具有非难等分区。

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