Linear Algebra and its Applications最新文献

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On the maximal ranks of some third-order quaternion tensors

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-18 DOI: 10.1016/j.laa.2024.12.010

Y.G. Liang, Yang Zhang

A complex tensor T can be considered as a quaternion tensor. Consequently, decomposing T using simple quaternion tensors, rather than simple complex tensors, can potentially result in decompositions with a smaller rank. In this paper, we first present an example demonstrating this. Furthermore, we show that the maximal rank of a 3 × 3 × 3 quaternion tensor is 5, and in doing so provide explicit decompositions into simple tensors with several cases. Finally, we provide the maximal ranks for all

n_{1} \times n_{2} \times n_{3}

quaternion tensors with

2 \leq n_{1}, n_{2}, n_{3} \leq 3

引用次数: 0

Eigenvalues of parametric rank-one perturbations of matrix pencils

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-18 DOI: 10.1016/j.laa.2024.12.012

Hannes Gernandt , Carsten Trunk

We study the behavior of eigenvalues of regular matrix pencils under rank-one perturbations which depend on a scalar parameter. In particular, the change of the algebraic multiplicities, the change of the eigenvalues for small parameter variations, as well as the asymptotic eigenvalue behavior as the parameter tends to infinity, is described. Besides that, an interlacing result for rank-one perturbations of matrix pencils is obtained. Finally, we show how to use these results in the redesign of electrical circuits, like for the low pass filter or for a two-stage CMOS operational amplifier.

引用次数: 0

Extensions of Riccati diagonal stability

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-18 DOI: 10.1016/j.laa.2024.12.014

Ali Algefary , Jianhong Xu

As generalizations of Riccati diagonal stability on a matrix pair, the notions of Riccati α-scalar stability and α-diagonal stability are introduced and fully characterized. Further extensions involving different block diagonal structures, simultaneous α-scalar stability, and simultaneous α-diagonal stability are also presented.

引用次数: 0

A note on the positivity of inverse operators acting on C⋆-algebras

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-16 DOI: 10.1016/j.laa.2024.12.006

Jochen Glück , Ulrich Groh

For a positive and invertible linear operator T acting on a C^⋆-algebra, we give necessary and sufficient criteria for the inverse operator

T^{- 1}

to be positive, too. Moreover, a simple counterexample shows that

T^{- 1}

need not be positive even if T is unital and its spectrum is contained in the unit circle.

引用次数: 0

Smith forms of matrices in Companion Rings, with group theoretic and topological applications

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-16 DOI: 10.1016/j.laa.2024.12.003

Vanni Noferini , Gerald Williams

Let R be a commutative ring and

g (t) \in R [t]

a monic polynomial. The commutative ring of polynomials

f (C_{g})

in the companion matrix

C_{g}

g (t)

, where

f (t) \in R [t]

, is called the Companion Ring of

g (t)

. Special instances include the rings of circulant matrices, skew-circulant matrices, pseudo-circulant matrices, or lower triangular Toeplitz matrices. When R is an Elementary Divisor Domain, we develop new tools for computing the Smith forms of matrices in Companion Rings. In particular, we obtain a formula for the second last non-zero determinantal divisor, we provide an

f (C_{g}) \leftrightarrow g (C_{f})

swap theorem, and a composition theorem. When R is a principal ideal domain we also obtain a formula for the number of non-unit invariant factors. By applying these to families of circulant matrices that arise as relation matrices of cyclically presented groups, in many cases we compute the groups' abelianizations. When the group is the fundamental group of a three dimensional manifold, this provides the homology of the manifold. In other cases we obtain lower bounds for the rank of the abelianization and record consequences for finiteness or solvability of the group, or for the Heegaard genus of a corresponding manifold.

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引用次数: 0

The truncated univariate rational moment problem

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-13 DOI: 10.1016/j.laa.2024.12.009

Rajkamal Nailwal , Aljaž Zalar

Given a closed subset K in

R

, the rational K–truncated moment problem (K–RTMP) asks to characterize the existence of a positive Borel measure μ, supported on K, such that a linear functional

L

, defined on all rational functions of the form

\frac{f}{q}

, where q is a fixed polynomial with all real zeros of even order and f is any real polynomial of degree at most 2k, is an integration with respect to μ. The case of a compact set K was solved in [4], but there is no argument that ensures that μ vanishes on all real zeros of q. An obvious necessary condition for the solvability of the K–RTMP is that

L

is nonnegative on every f satisfying

f |_{K} \geq 0

. If

L

is strictly positive on every

0 \neq f |_{K} \geq 0

, we add the missing argument from [4] and also bound the number of atoms in a minimal representing measure. We show by an example that nonnegativity of

L

is not sufficient and add the missing conditions to the solution. We also solve the K–RTMP for unbounded K and derive the solutions to the strong truncated Hamburger moment problem and the truncated moment problem on the unit circle as special cases.

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引用次数: 0

On the complex stability radius for time-delay differential-algebraic systems

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-13 DOI: 10.1016/j.laa.2024.12.007

Alexander Malyshev , Miloud Sadkane

An algorithm is proposed for computing the complex stability radius of a linear differential-algebraic system with a single delay and including a neutral term. The exponential factor in the characteristic equation is replaced by its Padé approximant thus reducing the level set method for finding the stability radius to a rational matrix eigenvalue problem. The level set method is coupled with a quadratically convergent iteration. An important condition relating the algebraic constraint and neutral term is introduced to eliminate the presence of characteristic roots approaching the imaginary axis at infinity. The number of iterations of the algorithm is roughly proportional to the numerical value of this condition. Effectiveness of the algorithm is illustrated by numerical examples.

引用次数: 0

The maximum spectral radius of P2k+1-free graphs of given size

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-12 DOI: 10.1016/j.laa.2024.12.008

Benju Wang, Bing Wang

<div><div>In this paper, we consider a Brualdi-Hoffman-Turán problem for graphs without path of given length. Denote by <math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math> the graph obtained from a cycle <math><msub><mrow><mi>C</mi></mrow><mrow><mi>t</mi></mrow></msub></math> by linking two vertices of distance two in the cycle. Recently, Li, Zhai and Shu showed that for <math><mi>k</mi><mo>≥</mo><mn>3</mn></math> and a graph G of size <math><mi>m</mi><mo>≥</mo><mn>16</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math>, if G is <math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math>-free or <math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math>-free, then the maximum adjacency spectral radius <math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt><mo>)</mo></math>. It follows immediately that if G is <math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math>-free of size <math><mi>m</mi><mo>≥</mo><mn>16</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math>, then <math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt><mo>)</mo></math>. However, the upper bound is not sharp. We consider the case for <math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math>-free graphs and obtain the following result: Let <math><mi>k</mi><mo>≥</mo><mn>4</mn></math> and G be a <math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math>-free graph of size <math><mi>m</mi><mo>≥</mo><mn>4</mn><msup><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup></math>. Then <math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>

引用次数: 0

Row or column completion of polynomial matrices of given degree II

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-10 DOI: 10.1016/j.laa.2024.12.004

Agurtzane Amparan , Itziar Baragaña , Silvia Marcaida , Alicia Roca

The row (column) completion problem of polynomial matrices of given degree with prescribed eigenstructure has been studied in [1], where several results of prescription of some of the four types of invariants that form the eigenstructure have also been obtained. In this paper we conclude the study, solving the completion for the cases not covered there. More precisely, we solve the row completion problem of a polynomial matrix when we prescribe the infinite (finite) structure and column and/or row minimal indices, and finally the column and/or row minimal indices. The necessity of the characterizations obtained holds to be true over arbitrary fields in all cases, whilst to prove the sufficiency it is required, in some of the cases, to work over algebraically closed fields.

引用次数: 0

Some observations on Erdős matrices

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-12-10 DOI: 10.1016/j.laa.2024.12.002

Raghavendra Tripathi

In a seminal paper in 1959, Marcus and Ree proved that every

n \times n

bistochastic matrix A satisfies

{‖ A ‖}_{F}^{2} \leq \max_{σ \in S_{n}} A_{i, σ (i)}

where

S_{n}

is the symmetric group on

{1, \dots, n}

. Erdős asked to characterize the bistochastic matrices for which the equality holds in the Marcus–Ree inequality. We refer to such matrices as Erdős matrices. While this problem is trivial in dimension

n = 2

, the case of dimension

n = 3

was only resolved recently in [4] in 2023. We prove that for every n, there are only finitely many

n \times n

Erdős matrices. We also give a complete characterization of Erdős matrices that yields an algorithm to generate all Erdős matrices in any given dimension. We also prove that Erdős matrices can have only rational entries. This answers a question of [4].

{"title":"Some observations on Erdős matrices","authors":"Raghavendra Tripathi","doi":"10.1016/j.laa.2024.12.002","DOIUrl":"10.1016/j.laa.2024.12.002","url":null,"abstract":"<div><div>In a seminal paper in 1959, Marcus and Ree proved that every <math><mi>n</mi><mo>×</mo><mi>n</mi></math> bistochastic matrix A satisfies <math><msubsup><mrow><mo>‖</mo><mi>A</mi><mo>‖</mo></mrow><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≤</mo><msub><mrow><mi>max</mi></mrow><mrow><mi>σ</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo>⁡</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>σ</mi><mo>(</mo><mi>i</mi><mo>)</mo></mrow></msub></math> where <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is the symmetric group on <math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math>. Erdős asked to characterize the bistochastic matrices for which the equality holds in the Marcus–Ree inequality. We refer to such matrices as Erdős matrices. While this problem is trivial in dimension <math><mi>n</mi><mo>=</mo><mn>2</mn></math>, the case of dimension <math><mi>n</mi><mo>=</mo><mn>3</mn></math> was only resolved recently in [4] in 2023. We prove that for every n, there are only finitely many <math><mi>n</mi><mo>×</mo><mi>n</mi></math> Erdős matrices. We also give a complete characterization of Erdős matrices that yields an algorithm to generate all Erdős matrices in any given dimension. We also prove that Erdős matrices can have only rational entries. This answers a question of [4].</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 236-251"},"PeriodicalIF":1.0,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

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