Linear Algebra and its Applications最新文献

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Spectral extremal graphs for disjoint odd wheels

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.034

Yu Luo , Zhenyu Ni , Yanxia Dong

For a given graph F, let

ex (n, F)

and

spex (n, F)

be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all F-free graphs of order n, respectively.

EX (n, F)

and

SPEX (n, F)

consist of the extremal graphs associated with

ex (n, F)

and

spex (n, F)

, respectively. The odd wheel

W_{2 k + 1}

is constructed by joining a vertex to a cycle

C_{2 k}

. Cioabă, Desai and Tait determined the spectral extremal graphs of

W_{2 k + 1}

for

k \geq 2, k \notin {4, 5}

. Xiao and Zamora determined the Turán number and all extremal graphs for

t W_{2 k + 1}

, which is the union of t vertex-disjoint copies of

W_{2 k + 1}

for

k \geq 3

. In this paper, we focus on the graph with maximum spectral radius among those that exclude any subgraph isomorphic to

t W_{2 k + 1}

. We present structural characteristics of these spectral extremal graphs for

k \geq 3, k \notin {4, 5}

. Furthermore, we demonstrate that

SPEX (n, t W_{2 k + 1}) \cap EX (n, t W_{2 k + 1}) = \emptyset

for

k \geq 10

and n large enough.

引用次数: 0

Constructing adjacency and distance cospectral graphs via regular rational orthogonal matrix

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.033

Lihuan Mao , Yuanhang Xu , Fenjin Liu , Bei Liu

Two graphs G and H are cospectral if they share the same spectrum. Constructing cospectral non-isomorphic graphs has been studied extensively for many years and various constructions are known in the literature. In this paper, we construct infinite families of adjacency cospectral graphs through the GM-switching method based on generalized Johnson graphs. We give some graph operations (e.g. rooted-product, corona, cartesian product, and coalescence) to construct distance cospectral graphs with different edges via a regular rational orthogonal matrix.

引用次数: 0

Polynomial codimension growth of superalgebras with superautomorphism

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.laa.2025.01.035

Sara Accomando

In this paper we present some results concerning associative superalgebras endowed with a superautomorphism of order ≤2. We characterize the superalgebras with superautomorphism with multiplicities of the cocharacter bounded by a constant. Moreover, we determine the characterization of the superalgebras with superautomorphism with polynomial growth of the codimensions and we give a classification of the subvarieties of the varieties of almost polynomial growth. Finally, we characterize the superalgebras with superautomorphism with linear growth of the codimensions.

引用次数: 0

The Rao-Mitra-Bhimasankaram relation is strongly antisymmetric

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-27 DOI: 10.1016/j.laa.2025.01.029

Oskar Maria Baksalary , Dennis Bernstein

For

n \times m

complex matrices A and B, the Rao-Mitra-Bhimasankaram (RMB) relation

\overset{RMB}{\leq}

, defined by

A \overset{RMB}{\leq} B

A A^{⁎} A = A B^{⁎} A

, is reflexive and antisymmetric, but not transitive. This paper shows that, despite the lack of transitivity,

\overset{RMB}{\leq}

is strongly antisymmetric in the sense that, for all integers

n \geq 2

A_{1} \overset{RMB}{\leq} \dots \overset{RMB}{\leq} A_{n} \overset{RMB}{\leq} A_{1}

implies

A_{1} = \dots = A_{n}

. The proof of this result is based on a novel proof that

\overset{RMB}{\leq}

is antisymmetric.

引用次数: 0

Rosenbrock's theorem on system matrices over elementary divisor domains

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-23 DOI: 10.1016/j.laa.2025.01.028

Froilán M. Dopico , Vanni Noferini , Ion Zaballa

Rosenbrock's theorem on polynomial system matrices is a classical result in linear systems theory that relates the Smith-McMillan form of a rational matrix G with the Smith form of an irreducible polynomial system matrix P giving rise to G and the Smith form of a submatrix of P. This theorem has been essential in the development of algorithms for computing the poles and zeros of a rational matrix via linearizations and generalized eigenvalue algorithms. In this paper, we extend Rosenbrock's theorem to system matrices P with entries in an arbitrary elementary divisor domain

R

and matrices G with entries in the field of fractions of

R

. These are the most general rings where the involved Smith-McMillan and Smith forms both exist and, so, where the problem makes sense. Moreover, we analyze in detail what happens when the system matrix is not irreducible. Finally, we explore how Rosenbrock's theorem can be extended when the system matrix P itself has entries in the field of fractions of the elementary divisor domain.

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引用次数: 0

When does subtracting a rank-one approximation decrease tensor rank?

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-23 DOI: 10.1016/j.laa.2025.01.025

Emil Horobeţ , Ettore Teixeira Turatti

Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for which subtracting some of its critical rank-one approximation from it and repeating the process we will eventually get to zero? In this article, we show how to construct this variety of tensors and we show how this is connected to the bottleneck points of the variety of rank-one tensors (and in general to the singular locus of the hyperdeterminant), and how this variety can be equal to and in some cases be more than (weakly) orthogonally decomposable tensors.

引用次数: 0

On the nullity of middle graphs

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-23 DOI: 10.1016/j.laa.2025.01.030

Xinmei Yuan , Danyi Li , Weigen Yan

Let G be a connected graph, and let

L (G)

and

M (G)

be the line graph and middle graph of G. Gutman and Sciriha (On the nullity of line graphs of trees, Discrete Mathematics, 232 (2001), 35-45) proved that the nullity

η (L (T))

L (T)

of a tree T satisfies

η (L (T)) = 0

η (L (T)) = 1

. But the problem to determine which trees T satisfy

η (L (T)) = 0

η (L (T)) = 1

is still open. In this paper, we prove that

η (M (G)) = 1

if G is a bipartite graph, and

η (M (G)) = 0

otherwise. As an application, we show that

η (G (n, m)) = 1

for the so-called silicate network

G (n, m)

obtained from the hexagonal lattice in the context of statistical physics.

引用次数: 0

The dimension formula for certain twisted Jacquet modules of a cuspidal representation of GL(n,Fq)

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-22 DOI: 10.1016/j.laa.2025.01.027

Kumar Balasubramanian , Himanshi Khurana

Let

n \geq 2

be a positive integer. Let F be the finite field of order q and

G = GL (n, F)

. Let

P = M N

be the standard parabolic subgroup of G corresponding to the partition

(k, n - k)

. Let

A \in M ((n - k) \times k, F)

be a rank t matrix. In this paper, we compute the dimension formula for the twisted Jacquet module

π_{N, ψ_{A}}

that depends on

n, k

and t, when π is an irreducible cuspidal representation of G and

ψ_{A}

is a character of N associated with A.

引用次数: 0

Multiple typical ranks in matrix completion

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-21 DOI: 10.1016/j.laa.2025.01.026

Mareike Dressler , Robert Krone

Low-rank matrix completion addresses the problem of completing a matrix from a certain set of generic specified entries. Over the complex numbers a matrix with a given entry pattern can be uniquely completed to a specific rank, called the generic completion rank. Completions over the reals may generically have multiple completion ranks, called typical ranks. We demonstrate techniques for proving that many sets of specified entries have only one typical rank, and show other families with two typical ranks, specifically focusing on entry sets represented by circulant graphs. This generalizes the results of Bernstein, Blekherman, and Sinn. In particular, we provide a complete characterization of the set of unspecified entries of an

n \times n

matrix such that

n - 1

is a typical rank and fully determine the typical ranks of an

n \times n

matrix with unspecified diagonal for

n < 9

. Moreover, we study the asymptotic behavior of typical ranks and present results regarding unique matrix completions.

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引用次数: 0

Linear preservers of parallel matrix pairs with respect to the k-numerical radius

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-21 DOI: 10.1016/j.laa.2025.01.019

Bojan Kuzma , Chi-Kwong Li , Edward Poon , Sushil Singla

Let

1 \leq k < n

be integers. Two

n \times n

matrices A and B form a parallel pair with respect to the k-numerical radius

w_{k}

w_{k} (A + μ B) = w_{k} (A) + w_{k} (B)

for some scalar μ with

| μ | = 1

; they form a TEA (triangle equality attaining) pair if the preceding equation holds for

μ = 1

. We classify linear bijections on

M_{n}

and on

H_{n}

which preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of

w_{k}

-isometries, except for some exceptional maps on

H_{n}

when

n = 2 k

引用次数: 0

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