Linear Algebra and its Applications最新文献

英文中文

Curves and spectrum localization for real matrices 实矩阵的曲线与谱定位

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-16 DOI: 10.1016/j.laa.2025.12.013

Aikaterini Aretaki , Maria Adam , Michael Tsatsomeros

It is well known that the eigenvalues of a complex matrix A are located to the left of the vertical line passing through the largest eigenvalue of its Hermitian part,

H (A)

. Adam and Tsatsomeros in [1] defined a cubic algebraic curve, known as the shell

Γ_{1} (A)

of A, using the two largest eigenvalues of

H (A)

. This curve localizes the spectrum further and lies to the left of the aforementioned vertical line. Later, Bergqvist in [5] extended the methodology employed in [1] to define a new curve,

Γ_{2} (A)

, in terms of the three largest eigenvalues of

H (A)

. This article delves into the geometry of

Γ_{2} (A)

for a real matrix A to address some open questions raised in [5]. In particular, specific conditions are established to characterize the configurations of

Γ_{2} (A)

in certain cases. Additionally, the number of eigenvalues of A surrounded by a bounded branch of the curve is examined. Examples are used to validate our findings and demonstrate the quality of

Γ_{2} (A)

as a finer spectrum localization area when compared to

Γ_{1} (A)

众所周知，复矩阵a的特征值位于穿过其厄米部分最大特征值H(a)的垂直线的左侧。Adam和Tsatsomeros在[1]中使用H(a)的两个最大特征值定义了a的三次代数曲线Γ1(a)。这条曲线进一步定位了光谱，位于前面提到的垂直线的左边。后来，Bergqvist在[5]中扩展了[1]中使用的方法，根据H(a)的三个最大特征值定义了一条新曲线Γ2(a)。本文将深入研究实数矩阵A的Γ2(A)的几何结构，以解决[5]中提出的一些开放式问题。特别地，建立了特定条件来表征Γ2(A)在某些情况下的构型。此外，还研究了A被曲线的有界分支所包围的特征值的个数。示例用于验证我们的发现，并证明与Γ1(A)相比，Γ2(A)作为更精细的光谱定位区域的质量。

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引用次数: 0

Jordan homomorphisms of triangular algebras over noncommutative algebras 非交换代数上三角代数的约当同态

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-15 DOI: 10.1016/j.laa.2025.12.011

Oksana Bezushchak

D. Benkovič described Jordan homomorphisms of algebras of triangular matrices over a commutative unital ring without additive 2-torsion. We extend this result to the case of noncommutative rings and remove the assumption of additive torsion.

Let R be an associative unital algebra over a commutative unital ring Φ. Consider the algebra

T_{n} (R)

of triangular

n \times n

matrices over R, and its subalgebra

T_{n}^{0} (R)

consisting of matrices whose main diagonal entries lie in Φ. We prove that for any Jordan homomorphism of

T_{n} (R)

, its restriction to

T_{n}^{0} (R)

is standard.

描述了无加性2-扭转的可交换一元环上三角矩阵代数的约当同态。我们将这个结果推广到非交换环的情况，去掉了加性扭转的假设。设R是可交换单环上的一个结合单代数Φ。考虑R上的三角形n×n矩阵的代数Tn(R)及其子代数Tn0(R)，它由主要对角线元素位于Φ的矩阵组成。证明了对于Tn(R)的任意Jordan同态，其对Tn0(R)的限制是标准的。

引用次数: 0

Matrix best approximation in the spectral norm 谱范数中矩阵的最佳逼近

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-15 DOI: 10.1016/j.laa.2025.12.007

Vance Faber , Jörg Liesen , Petr Tichý

We derive, similar to Lau and Riha in [22], a matrix formulation of a general best approximation theorem of Singer for the special case of spectral approximations of a given matrix from a given subspace. Using our matrix formulation we describe the relation of the spectral approximation problem to semidefinite programming, and we present a simple MATLAB code to solve the problem numerically. We then obtain geometric characterizations of spectral approximations that are based on the k-dimensional field of k matrices, which we illustrate with several numerical examples. The general spectral approximation problem is a min-max problem, whose value is bounded from below by the corresponding max-min problem. Using our geometric characterizations of spectral approximations, we derive several necessary and sufficient as well as sufficient conditions for equality of the max-min and min-max values. Finally, we prove that the max-min and min-max values are always equal for

2 \times 2

block diagonal matrices containing two identical diagonal blocks. Several results in this paper generalize results that have been obtained in the convergence analysis of the GMRES method for solving linear algebraic systems.

与[22]中的Lau和Riha相似，对于给定子空间中给定矩阵的谱近似的特殊情况，我们导出了Singer的一般最佳近似定理的矩阵形式。利用矩阵公式描述了谱逼近问题与半定规划问题的关系，并给出了一个简单的MATLAB程序来进行数值求解。然后，我们得到基于k矩阵的k维域的谱近似的几何特征，我们用几个数值例子来说明。一般的谱逼近问题是一个最小-最大问题，其值由相应的最大-最小问题下界。利用谱近似的几何特征，导出了最大-最小值和最小-最大值相等的几个充分必要条件。最后，我们证明了含有两个相同对角块的2×2块对角矩阵的max-min和min-max值总是相等的。本文的几个结果推广了GMRES法求解线性代数系统的收敛性分析中所得到的结果。

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引用次数: 0

A sharp spectral extremal result for general non-bipartite graphs 一般非二部图的尖锐谱极值结果

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-15 DOI: 10.1016/j.laa.2025.12.010

John Byrne

For a graph family

F

, let

ex (n, F)

and

spex (n, F)

denote the maximum number of edges and maximum spectral radius of an n-vertex

F

-free graph, respectively, and let

EX (n, F)

and

SPEX (n, F)

denote the corresponding sets of extremal graphs. Wang, Kang, and Xue showed that if

r \geq 2

and

ex (n, F) = e (T_{n, r}) + O (1)

then

SPEX (n, F) \subseteq EX (n, F)

for n large enough. Fang, Tait, and Zhai extended this result by showing if

e (T_{n, r}) \leq ex (n, F) < e (T_{n, r}) + ⌊ n / 2 r ⌋

then

SPEX (n, F) \subseteq EX (n, F)

for n large enough, and asked for the maximum constant

c (r)

such that

ex (n, F) \leq e (T_{n, r}) + (c (r) - ε) n

guarantees such containment. In this paper we determine

c (r)

exactly for all

r \geq 3

对于图族F，设ex（n，F）和spex（n，F）分别表示有n顶点的无F图的最大边数和最大谱半径，设ex（n，F）和spex（n，F）表示相应的极值图集。Wang、Kang、Xue证明了当r≥2且ex(n，F)=e(Tn,r)+O(1)时，当n足够大时，SPEX(n，F)≥exp （n，F）。Fang、Tait和Zhai对这一结果进行了推广，证明如果e(Tn,r)≤ex(n，F)<e(Tn,r)+⌊n/2r⌋，则SPEX（n，F）对n足够大，并要求最大常数c(r)使得ex(n，F)≤e(Tn,r)+(c(r)−ε)n保证了该包含性。在本文中，我们精确地确定了所有r≥3的c(r)。

引用次数: 0

Preserving Lefschetz properties after extension of variables 在扩展变量后保留Lefschetz属性

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-12 DOI: 10.1016/j.laa.2025.12.008

Filip Jonsson Kling

Consider a standard graded artinian k-algebra B and an extension of B by a new variable,

A = B \otimes_{k} k [x] / (x^{d})

for some

d \geq 1

. We will show how maximal rank properties for powers of a general linear form on A can be determined by maximal rank properties for different powers of general linear forms on B. This is then used to study Lefschetz properties of algebras that can be obtained via such extensions. In particular, it allows for a new proof that monomial complete intersections have the strong Lefschetz property over a field of characteristic zero. Moreover, it gives a recursive formula for the determinants that show up in that case. Finally, for algebras over a field of characteristic zero, we give a classification for what properties B must have for all extensions

B \otimes_{k} k [x] / (x^{d})

to have the weak or the strong Lefschetz property.

考虑一个标准的分级人工k-代数B和一个新变量对B的扩展，对于某些d≥1，a =B⊗kk[x]/(xd)。我们将展示a上一般线性形式的幂的极大秩性质如何由b上一般线性形式的不同幂的极大秩性质决定，然后用于研究通过这种扩展可以获得的代数的Lefschetz性质。特别地，它允许一个新的证明，证明单项式完全交在特征为零的域上具有强Lefschetz性质。而且，它给出了在这种情况下出现的行列式的递归公式。最后，对于特征为0的域上的代数，我们给出了B对于所有扩展B⊗kk[x]/(xd)具有弱或强Lefschetz性质所必须具有的性质的分类。

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引用次数: 0

Matrices with all diagonal entries lying on the boundary of the numerical range 所有对角线元素位于数值范围边界上的矩阵

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-08 DOI: 10.1016/j.laa.2025.12.005

Hwa-Long Gau , Chi-Kwong Li , Kuo-Zhong Wang

For an

n \times n

complex matrix A, we study the value

k (A)

, which is the maximum size of an orthonormal set

{x_{1}, \dots, x_{k}}

such that

x_{j}^{⁎} A x_{j}

lie on the boundary of

W (A)

for

j = 1, \dots, k

. We give a complete characterization of matrices A with

k (A) = n

, and determine when such a matrix has reducing subspaces. Furthermore, we characterize companion matrices and nonnegative upper triangular the Toeplitz matrices A with

k (A) = n

对于一个n×n复矩阵A，我们研究了值k(A)，它是一个标准正交集合{x1，…，xk}的最大大小，使得xj Axj位于W(A)的边界上，当j=1，…，k时。给出了矩阵a在k(a)=n时的完备刻画，并确定了这种矩阵何时具有约简子空间。进一步，我们刻画了k(A)=n的伴矩阵和非负上三角形的Toeplitz矩阵A。

引用次数: 0

Comparing the operator norms of symmetric matrices sharing the same numerical range 比较具有相同数值范围的对称矩阵的算子范数

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-05 DOI: 10.1016/j.laa.2025.12.004

Mao-Ting Chien , Hiroshi Nakazato

The ternary form of an

n \times n

matrix A is defined by

F_{A} (t, x, y) = \det (t I_{n} + x ℜ (A) + y ℑ (A))

, where

ℜ (A) = (A + A^{⁎}) / 2

and

ℑ (A) = (A - A^{⁎}) / (2 i)

. If the algebraic curve

F_{A} (t, x, y) = 0

has no singular points, the Helton-Vinnikov theorem asserts that there are

2^{g}

non-unitarily similar symmetric matrices S satisfying

F_{A} (t, x, y) = F_{S} (t, x, y)

, where

g = (n - 1) (n - 2) / 2

. We compare the operator norms of the

2^{g}

symmetric matrices that share the same numerical range of A.

三元形式的一个n×n矩阵被定义为FA (t, x, y) =检波器(锡+ xℜ(A) + yℑ(A)),在ℜ(A) = (A +⁎)/ 2和ℑ(A) =(−⁎)/(2)。如果代数曲线FA(t,x,y)=0没有奇点，则Helton-Vinnikov定理断言存在2g个非酉相似对称矩阵S满足FA(t,x,y)=FS(t,x,y)，其中g=(n−1)(n−2)/2。我们比较了具有相同数值范围A的2g对称矩阵的算子范数。

引用次数: 0

Covariant decomposable maps on C*-algebras and quantum dynamics C*-代数上的协变可分解映射与量子动力学

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-04 DOI: 10.1016/j.laa.2025.12.002

Krzysztof Szczygielski

We characterize covariant positive decomposable maps between unital C*-algebras in terms of a dilation theorem, which generalizes a seminal result by H. Scutaru from (1979) [7]. As a case study, we provide a certain characterization of the operator sum representation of maps on

M_{n} (C)

, covariant with respect to the maximal commutative subgroup of

U (n)

. A connection to quantum dynamics is established by specifying sufficient and necessary conditions for covariance of D-divisible (decomposably divisible) quantum evolution families, recently introduced in Szczygielski (2023) [11].

利用膨胀定理刻画了单位C*-代数间的协变正可分解映射，该定理推广了H. Scutaru(1979)[7]的一个重要结果。作为一个案例研究，我们给出了Mn(C)上映射的算子和表示的一个特征，它相对于U(n)的最大交换子群是协变的。通过指定d可分（可分解可分）量子演化族协方差的充分必要条件，建立了与量子动力学的联系，最近在Szczygielski(2023)[11]中介绍。

引用次数: 0

Colorful positive bases decomposition and Helly-type results for cones 锥体的彩色正基分解和helly型结果

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-03 DOI: 10.1016/j.laa.2025.11.023

Grigory Ivanov

We prove the following colorful Helly-type result: Fix

k \in [d - 1]

. Assume

A_{1}, \dots, A_{d + (d - k) + 1}

are finite sets (colors) of nonzero vectors in

R^{d}

. If for every rainbow sub-selection R from these sets of size at most

\max {d + 1, 2 (d - k + 1)}

, the system

〈 a, x 〉 \leq 0, a \in R

has at least k linearly independent solutions, then at least one of the systems

〈 a, x 〉 \leq 0, a \in A_{i}

i \in [d + (d - k) + 1]

has at least k linearly independent solutions.

A rainbow sub-selection from several sets refers to choosing at most one element from each set (color).

The Helly number

\max {d + 1, 2 (d - k + 1)}

and the number of colors

d + (d - k) + 1

are optimal.

Our key observation is a certain colorful Carathéodory-type result for positive bases.

我们证明了以下多彩的helly型结果：固定k∈[d−1]。假设A1，…，Ad+(d−k)+1是Rd中非零向量的有限集合（颜色）。如果对于这些集合中大小不超过max (d +1,2(d−k+1)}的每个彩虹子选择R，系统< a,x >≤0，a∈R至少有k个线性无关的解，那么系统< a,x >≤0,a∈Ai, i∈[d+(d−k)+1]中至少有一个系统< a,x >≤0,a∈Ai, i∈[d+(d−k)+1]至少有k个线性无关的解。从几个集合中选择彩虹子是指从每个集合中最多选择一个元素（颜色）。Helly数max (d +1,2(d−k+1)}和颜色数d+(d−k)+1是最优的。我们的主要观察结果是阳性碱基的某种彩色carathacemory型结果。

引用次数: 0

Every expansive m-concave operator has m-isometric dilation 每一个膨胀m-凹算子都有m-等距膨胀

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-03 DOI: 10.1016/j.laa.2025.12.001

Michał Buchała

The aim of this paper is to obtain m-isometric dilation of expansive m-concave operator on Hilbert space. The obtained dilation is shown to be minimal. The matrix representation of this dilation is given. It is also proved that in case of 3-concave operators the assumption on expansivity is not necessary. The paper contains an example showing that minimal m-isometric dilations may not be isomorphic.

本文的目的是得到Hilbert空间上膨胀m-凹算子的m等距扩张。得到的膨胀是最小的。给出了这种膨胀的矩阵表示。同时证明了对于3凹算子，不需要膨胀性的假设。本文给出了一个例子，证明最小m-等距膨胀可能不是同构的。

引用次数: 0

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Linear Algebra and its Applications

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