Linear Algebra and its Applications最新文献

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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

On the loss of orthogonality in low-synchronization variants of reorthogonalized block classical Gram-Schmidt 重正交化块的低同步变体的正交性损失

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-02 DOI: 10.1016/j.laa.2025.11.018

Erin Carson , Kathryn Lund , Yuxin Ma , Eda Oktay

Interest in communication-avoiding orthogonalization schemes for high-performance computing has been growing recently. This manuscript addresses open questions about the numerical stability of various block classical Gram-Schmidt variants that have been proposed in the past few years. An abstract framework is employed, the flexibility of which allows for new rigorous bounds on the loss of orthogonality in these variants. We first analyze a generalization of (reorthogonalized) block classical Gram-Schmidt and show that a “strong” intrablock orthogonalization routine is only needed for the very first block in order to maintain orthogonality on the level of the unit roundoff. In particular, this “strong” first step does not have to be a reorthogonalized QR itself and subsequent steps can use less stable QR variants, thus keeping the overall communication costs low.

Then, using this variant, which has four synchronization points per block column, we remove the synchronization points one at a time and analyze how each alteration affects the stability of the resulting method. Our analysis shows that the variant requiring only one synchronization per block column, equivalent to a variant previously proposed in the literature, cannot be guaranteed to be stable in practice, as stability begins to degrade with the first reduction of synchronization points. As a negative result, we conclude that this particular block algorithm should be avoided in practice.

Our analysis of block methods also provides new, more positive theoretical results for the single-column case. In particular, it is proven that DCGS2 from (Bielich et al., 2022 [5]) and CGS-2 from (Świrydowicz et al., 2021 [10]) are as stable as Householder QR. Numerical examples from the BlockStab toolbox are included throughout, to help compare variants and illustrate the effects of different choices of intraorthogonalization subroutines.

最近，人们对用于高性能计算的避免通信的正交化方案越来越感兴趣。这篇手稿解决了关于过去几年提出的各种块经典Gram-Schmidt变异体的数值稳定性的开放问题。采用抽象框架，其灵活性允许在这些变体中对正交性损失的新的严格界限。我们首先分析了（重新正交化）块经典Gram-Schmidt的推广，并表明为了在单位舍入水平上保持正交性，只需要在第一个块上使用“强”块内正交化例程。特别是，这个“强”的第一步不必是重新正交化的QR本身，后续步骤可以使用不太稳定的QR变体，从而保持整体通信成本较低。然后，使用这种每个块列有四个同步点的变体，我们一次删除一个同步点，并分析每次更改如何影响结果方法的稳定性。我们的分析表明，每个块列只需要一个同步的变体，相当于以前在文献中提出的变体，不能保证在实践中是稳定的，因为随着同步点的首次减少，稳定性开始下降。作为否定的结果，我们得出结论，在实践中应该避免这种特殊的块算法。我们对块方法的分析也为单列情况提供了新的、更积极的理论结果。特别是证明了（Bielich et al., 2022[5]）中的DCGS2和（Świrydowicz et al., 2021[10]）中的CGS-2与Householder QR一样稳定。从BlockStab工具箱中的数值示例包括在整个过程中，以帮助比较变量，并说明不同选择的内正交化子例程的影响。

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引用次数: 0

Dilating contractions into involutions and projections 将宫缩扩张为内联和凸出

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-12-01 DOI: 10.1016/j.laa.2025.11.022

Jean-Christophe Bourin , Eun-Young Lee

Any contraction

A \in M_{n}

can be dilated (1) into an involution

S \in M_{2 n}

with operator norm

{‖ S ‖}_{\infty} \leq 1 + \sqrt{2}

and (2) into a projection

E \in M_{2 n}

with

{‖ E ‖}_{\infty} \leq 3

. The bounds

1 + \sqrt{2}

and 3 are the smallest possible ones.

任何收缩A∈Mn都可以展开(1)成运算范数为‖S‖∞≤1+2的对合S∈M2n，(2)成运算范数为‖E‖∞≤3的投影E∈M2n。1+2和3是最小的边界。

引用次数: 0

A note on almost cospectrality of components of NEPS of bipartite graphs 二部图的NEPS分量的几乎同谱性的注记

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-11-27 DOI: 10.1016/j.laa.2025.11.019

Ivan Stanković

For

B \subset {0, 1}^{n} - {(0, \dots, 0)}

and

S \subset {1, \dots, n}

, let

Ann (B, S) = {β \in B : (\forall i \in S) β_{i} = 0}

. Considering

B

also as a binary matrix with

| B |

rows and n columns, let

r (B)

denote the binary rank of

B

. We disprove here the conjecture of Stevanović [Linear Algebra Appl. 311 (2000) 35–44] that the components of NEPS of connected, bipartite graphs are almost cospectral whenever the basis

B

of NEPS satisfies the condition

Ann (B, S) \neq \emptyset \Rightarrow | S | + r (Ann (B, S)) \leq r (B)

B⊂{0,1}n−{(0。0)}和S⊂{1,…,n},让安(B S) ={β∈B:我(∀我∈S)β= 0}。考虑B也是一个有|B|行n列的二进矩阵，设r(B)表示B的二进秩。本文证明了stevanovic[线性代数应用311(2000)35-44]的猜想，即当连通二部图的NEPS的基B满足条件Ann(B，S)≠∅⇒|S|+r(Ann(B，S))≤r(B)时，NEPS的分量几乎是共谱的。

引用次数: 0

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