Linear Algebra and its Applications最新文献

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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

Total trades, intersection matrices and Specht modules 总交易，交叉矩阵和Specht模块

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-11-27 DOI: 10.1016/j.laa.2025.11.021

Mihalis Maliakas, Dimitra-Dionysia Stergiopoulou

Trades are important objects in combinatorial design theory that may be realized as certain elements of kernels of inclusion matrices. Total trades were introduced recently by Ghorbani, Kamali and Khosravshahi who showed that over a field of characteristic zero the vector space of trades decomposes into a direct sum of spaces of total trades. In this paper, we show that the vector space spanned by the permutations of a total trade is an irreducible representation of the symmetric group. As a corollary, the previous decomposition theorem is recovered. Also, a basis is obtained for the module of total trades in the spirit of Specht polynomials. In the second part of the paper we consider more generally intersection matrices and determine the irreducible decompositions of their images. This generalizes previously known results concerning ranks of special cases.

行业是组合设计理论中的重要对象，可以通过包含矩阵核的某些元素来实现。总交易量是最近由Ghorbani， Kamali和Khosravshahi引入的，他们证明了在特征为零的域上，交易量的向量空间分解为总交易量空间的直接和。在本文中，我们证明了由总交易的排列所张成的向量空间是对称群的不可约表示。作为一个推论，恢复了先前的分解定理。同时，根据Specht多项式的思想，得到了总交易量模块的一个基。在论文的第二部分，我们考虑了更一般的相交矩阵，并确定了其图像的不可约分解。这概括了以前已知的关于特殊情况等级的结果。

引用次数: 0

Gershgorin-type spectral inclusions for matrices 矩阵的gershgorin型光谱内含物

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-11-26 DOI: 10.1016/j.laa.2025.11.017

Simon N. Chandler-Wilde , Marko Lindner

In this paper we derive sequences of Gershgorin-type inclusion sets for the spectra and pseudospectra of finite matrices. In common with previous generalisations of the classical Gershgorin bound for the spectrum, our inclusion sets are based on a block decomposition. In contrast to previous generalisations that treat the matrix as a perturbation of a block-diagonal submatrix, our arguments treat the matrix as a perturbation of a block-tridiagonal matrix, which can lead to sharp spectral bounds, as we show for the example of large Toeplitz matrices. Our inclusion sets, which take the form of unions of pseudospectra of square or rectangular submatrices, build on our own recent work on inclusion sets for bi-infinite matrices in Chandler-Wilde et al. (2024) [3].

本文导出了有限矩阵谱和伪谱的gershgorin型包含集序列。与之前对频谱的经典Gershgorin界的推广一样，我们的包含集是基于块分解的。与先前将矩阵视为块对角子矩阵的扰动的推广相反，我们的论证将矩阵视为块三对角矩阵的扰动，这可能导致尖锐的谱界，正如我们为大型Toeplitz矩阵的示例所示。我们的包含集采用方形或矩形子矩阵的伪谱并集的形式，建立在我们自己最近在Chandler-Wilde等人（2024）[3]中对双无限矩阵的包含集的研究基础上。

引用次数: 0

Spectral radius, the matching number and fractional criticality of graphs 谱半径，匹配数和分数临界图

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-11-26 DOI: 10.1016/j.laa.2025.11.020

Huicai Jia , Ao Fan , Ruifang Liu

The binding number

b (G)

of a graph G is the minimum value of

| N_{G} (X) | / | X |

taken over all non-empty subsets X of

V (G)

such that

N_{G} (X) \neq V (G)

. A graph G is called 1-binding if

b (G) \geq 1

. The matching number of G, denoted by

ν (G)

, is the size of a maximum matching in G. In this paper, we focus on characterizing structural properties from the spectral perspective. Inspired by the elegant result of Fan and Lin (2024) [16] on the existence of a perfect matching in 1-binding graphs, we adopt the double eigenvector technique due to Rowlinson and present a tight sufficient condition in terms of the spectral radius for a connected 1-binding graph to guarantee

ν (G) > \frac{n - k}{2}

Let

r \geq 1

be a given integer. A graph G is fractional ID-

[a, b]

-factor-critical if for every independent set I of G with

| I | = r

G - I

has a fractional

[a, b]

-factor. We first propose a sufficient condition based on the number of edges for a graph to be fractional ID-

[a, b]

-factor-critical. As an application, we take full advantage of the spectral bound and obtain a sufficient condition in terms of the spectral radius for a graph to be fractional ID-

[a, b]

-factor-critical, which extends nicely the result of Fan et al. (2024) [12] from

a = b = 1

to general a and b.

图G的绑定数b(G)是|NG(X)|/|X|接管V(G)的所有非空子集X使NG(X)≠V(G)的最小值。若b(G)≥1，则图G为1绑定图。G的匹配数，用ν(G)表示，是G中最大匹配的大小。本文主要从光谱的角度来表征结构性质。受Fan and Lin(2024)[16]关于1-结合图存在完美匹配的优雅结果的启发，我们采用了Rowlinson的双特征向量技术，给出了连通1-结合图的谱半径保证ν(G)>；n−k2的严密充分条件。设r≥1是一个给定的整数。图G是分数阶ID-[A，b]-因子临界的，如果对于G的每一个独立集I (|I|=r)， G−I有分数阶[A，b]-因子。我们首先提出了一个基于边数的图为分数阶ID-[a，b]-因子临界的充分条件。作为应用，我们充分利用谱界，得到了谱半径为分数阶ID-[a，b]-因子临界的充分条件，很好地推广了Fan et al.(2024)[12]从a=b=1到一般a和b的结果。

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引用次数: 0

Estimating a matrix's singular values with interpolative decompositions 用插值分解估计矩阵的奇异值

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-11-21 DOI: 10.1016/j.laa.2025.11.013

Anil Damle , Silke Glas , Alex Townsend , Annan Yu

We study algorithms called rank-revealers that estimate a matrix's singular values by carefully selecting rows and/or columns. Such algorithms are fundamental in matrix compression, singular value estimation, and column subset selection problems. While column-pivoted QR has been widely adopted due to its practicality, it is not always a rank-revealer. Conversely, Gaussian elimination (GE) with global maximum volume (maxvol) pivoting is guaranteed to estimate a matrix's singular values but its exponential complexity limits its interest to theory. We show that the concept of near-local maxvol pivoting is a crucial and practical pivoting strategy for rank-revealers based on GE and QR. In particular, we prove that it is both necessary and sufficient; highlighting that all local solutions are nearly as good as the global one. This insight elevates Gu and Eisenstat's rank-revealing QR as an archetypal rank-revealer, and we implement a version that is observed to be at most 2× more computationally expensive than CPQR. We unify the landscape of rank-revealers by considering GE and QR together and prove that the success of any pivoting strategy can be assessed by its nearest to a local maxvol pivot.

我们研究了一种叫做秩揭示的算法，它通过仔细选择行和/或列来估计矩阵的奇异值。这些算法是矩阵压缩、奇异值估计和列子集选择问题的基础。虽然柱轴QR因其实用性而被广泛采用，但它并不总是一个显示排名的工具。相反，具有全局最大体积（maxvol）旋转的高斯消去（GE）保证估计矩阵的奇异值，但其指数复杂性限制了它的理论兴趣。我们证明了近局部最大卷旋转的概念是基于GE和QR的排名揭示器的关键和实用的旋转策略。特别地，我们证明了它既是必要的又是充分的；强调所有本地解决方案几乎与全球解决方案一样好。这一见解将Gu和Eisenstat的揭示等级的QR提升为一个原型等级揭示器，并且我们实现了一个版本，据观察，它的计算成本最多比CPQR高2倍。我们通过将GE和QR放在一起来统一排名揭示者的景观，并证明任何枢纽策略的成功都可以通过其最接近本地最大枢纽来评估。

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引用次数: 0

Generalizing Lee's conjecture on the sum of absolute values of matrices 推广李氏关于矩阵绝对值和的猜想

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-11-19 DOI: 10.1016/j.laa.2025.11.015

Quanyu Tang, Shu Zhang

<div><div>Let <math><msub><mrow><mo>‖</mo><mo>⋅</mo><mo>‖</mo></mrow><mrow><mi>p</mi></mrow></msub></math> denote the Schatten p-norm of matrices and <math><msub><mrow><mo>‖</mo><mo>⋅</mo><mo>‖</mo></mrow><mrow><mi>F</mi></mrow></msub></math> the Frobenius norm. For a square matrix X, let <math><mo>|</mo><mi>X</mi><mo>|</mo></math> denote its absolute value. In 2010, Eun-Young Lee posed the problem of determining the smallest constant <math><msub><mrow><mi>c</mi></mrow><mrow><mi>p</mi></mrow></msub></math> such that <math><msub><mrow><mo>‖</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo>‖</mo></mrow><mrow><mi>p</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>p</mi></mrow></msub><msub><mrow><mo>‖</mo><mspace></mspace><mo>|</mo><mi>A</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>B</mi><mo>|</mo><mspace></mspace><mo>‖</mo></mrow><mrow><mi>p</mi></mrow></msub></math> for all complex matrices <math><mi>A</mi><mo>,</mo><mi>B</mi></math>. The Frobenius case <math><mo>(</mo><mi>p</mi><mo>=</mo><mn>2</mn><mo>)</mo></math> conjectured by Lee was proved by Lin and Zhang (2022) [6] and re-proved by Zhang (2025) [7]. In this paper, we extend Lee's conjecture from two matrices to an arbitrary number <math><mi>m</mi><mo>≥</mo><mn>2</mn></math> of complex matrices <math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math>, and determine the sharp inequality<math><msub><mrow><mo>‖</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>‖</mo></mrow><mrow><mi>F</mi></mrow></msub><mo>≤</mo><msqrt><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mi>m</mi></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msqrt><mspace></mspace><msub><mrow><mo>‖</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>|</mo><mo>‖</mo></mrow><mrow><mi>F</mi></mrow></msub><mo>,</mo></math> with equality attained by an equiangular rank-one family. We further generalize Lee's problem by seeking the smallest constant <math><msub><mrow><mi>c</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>m</mi><mo>)</mo></math> such that <math><msub><mrow><mo>‖</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>‖</mo></mrow><mrow><mi>p</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>m</mi><mo>)</mo><mspace></mspace><msub><mrow><mo>‖</mo><m

设‖⋅‖p为矩阵的Schatten p范数，‖⋅‖F为Frobenius范数。对于一个方阵X，设|X|表示它的绝对值。2010年，Eun-Young Lee提出了对所有复矩阵A，B求最小常数cp的问题，使得‖A+B‖p≤cp‖|A|+|B|‖p。Lee猜想的Frobenius案例（p=2）被Lin和Zhang(2022)[6]证明，Zhang(2025)[7]重新证明。本文将李氏猜想从两个矩阵推广到复矩阵A1，…，Am的任意数m≥2，并确定了尖锐不等式‖∑k=1mAk‖F≤1+m2‖∑k=1m|Ak|‖F，该不等式由等角秩一族得到。我们进一步推广Lee的问题，寻求最小常数cp(m)使‖∑k=1mAk‖p≤cp(m)‖∑k=1m|Ak|‖p。证明了cp(m)≤(m)1−1/p，并推测了cp(m)的最优值的一个封闭表达式，该表达式可以恢复所有已知情况p=1,2,∞。

引用次数: 0

Adaptive coefficients iterative method for computing matrix inverse 自适应系数迭代法计算矩阵逆

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-11-19 DOI: 10.1016/j.laa.2025.11.016

Marko Kostadinov , Mihailo Krstić , Kostadin Rajković , Marko D. Petković

In this paper, we construct the new iterative method of the form

X_{k + 1} = X_{k} P_{k} (A X_{k})

where

P_{k}

is polynomial, for computing the inverse

A^{- 1}

of a given invertible matrix

A \in R^{n \times n}

. Coefficients of the polynomial

P_{k}

in k-th iteration are variable and determined in a way to minimize the Frobenius norm of the error matrix

I - A X_{k + 1}

. The convergence of the new method is investigated, where several theoretical results are proven. The method is compared to the existing iterative methods of the similar type, on a various numerical examples. The results show that the new method outperforms the existing ones for almost all test matrices. Moreover, they suggest that the new method posses almost global convergence.

本文构造了一种新的迭代方法，其形式为Xk+1=XkPk(AXk)，其中Pk为多项式，用于计算给定可逆矩阵A∈Rn×n的逆A−1。在第k次迭代中，多项式Pk的系数是可变的，并且以最小化误差矩阵I−AXk+1的Frobenius范数的方式确定。研究了新方法的收敛性，并证明了几个理论结果。通过各种数值算例，将该方法与现有的同类迭代方法进行了比较。结果表明，对于几乎所有的测试矩阵，新方法都优于现有的方法。此外，他们认为新方法几乎具有全局收敛性。

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引用次数: 0

A note on the quantum Wielandt inequality 关于量子维兰特不等式的注解

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-11-19 DOI: 10.1016/j.laa.2025.11.014

Owen Ekblad

In this note, we prove that the index of primitivity of any primitive unital Schwarz map is at most

2 {(D - 1)}^{2}

, where D is the dimension of the underlying matrix algebra. This inequality was first proved by Rahaman for Schwarz maps which were both unital and trace preserving. As we show, the assumption of unitality is basically innocuous, but in general not all primitive unital Schwarz maps are trace preserving. Therefore, the precise purpose of this note is to showcase how to apply the method of Rahaman to unital primitive Schwarz maps that don't preserve trace. As a corollary of this theorem, we show that the index of primitivity of any primitive 2-positive map is at most

2 {(D - 1)}^{2}

, so in particular this bound holds for arbitrary primitive completely positive maps. We briefly discuss of how this relates to a conjecture of Perez-Garcia, Verstraete, Wolf and Cirac.

在本文中，我们证明了任何原始一元Schwarz映射的基性指数不超过2(D−1)2，其中D是其基础矩阵代数的维数。这个不等式首先由Rahaman证明了Schwarz映射的单一性和迹迹保持性。正如我们所展示的，酉性假设基本上是无害的，但通常不是所有的原始酉Schwarz映射都是迹保持的。因此，本笔记的确切目的是展示如何将Rahaman方法应用于不保留迹的单原始Schwarz映射。作为该定理的一个推论，我们证明了任何原元2-正映射的原元索引不超过2(D−1)2，因此这个界特别适用于任意原元完全正映射。我们简要讨论了这与Perez-Garcia、Verstraete、Wolf和Cirac的猜想之间的关系。

引用次数: 0

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