Linear Algebra and its Applications最新文献

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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

Minimal error momentum Bregman-Kaczmarz

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-21 DOI: 10.1016/j.laa.2025.01.024

Dirk A. Lorenz, Maximilian Winkler

The Bregman-Kaczmarz method is an iterative method which can solve strongly convex problems with linear constraints and uses only one or a selected number of rows of the system matrix in each iteration, thereby making it amenable for large-scale systems. To speed up convergence, we investigate acceleration by heavy ball momentum in the so-called dual update. Heavy ball acceleration of the Kaczmarz method with constant parameters has turned out to be difficult to analyze, in particular no accelerated convergence for the

L_{2}

-error of the iterates has been proven to the best of our knowledge. Here we propose a way to adaptively choose the momentum parameter by a minimal-error principle similar to a recently proposed method for the standard randomized Kaczmarz method. The momentum parameter can be chosen to exactly minimize the error in the next iterate or to minimize a relaxed version of the minimal error principle. The former choice leads to a theoretically optimal step while the latter is cheaper to compute. We prove improved convergence results compared to the non-accelerated method. Numerical experiments show that the proposed methods can accelerate convergence in practice, also for matrices which arise from applications such as computational tomography.

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引用次数: 0

Maximum Aα-spectral radius of {C(3,3),C(4,3)}-free graphs

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-21 DOI: 10.1016/j.laa.2025.01.023

S. Pirzada, Amir Rehman

For a simple graph G and for any

α \in [0, 1]

, Nikiforov defined the generalized adjacency matrix as

A_{α} (G) = α D (G) + (1 - α) A (G)

, where

A (G)

and

D (G)

are the adjacency and degree diagonal matrices of G, respectively. The largest eigenvalue of

A_{α} (G)

is called the generalized adjacency spectral radius (or

A_{α}

-spectral radius) of G. Let

C (l, t)

denote the graph obtained from

C_{l}

and

C_{t}

by superimposing an edge of

C_{l}

with an edge of

C_{t}

. If a graph is free of both

C (3, 3)

and

C (4, 3)

, we call it a

{C (3, 3), C (4, 3)}

-free graph. In this paper, we give a sharp upper bound on the

A_{α}

-spectral radius of

{C (3, 3), C (4, 3)}

-free graphs for

α \in [0, \frac{1}{2})

. We show that the extremal graph attaining the bound is the 2-partite Turán graph.

引用次数: 0

Some families of digraphs determined by the complementarity spectrum

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-20 DOI: 10.1016/j.laa.2025.01.022

Diego Bravo , Florencia Cubría , Marcelo Fiori , Gustavo Rama

In this paper we study seven families of digraphs, and we determine whether digraphs in these families can be determined by their spectral radius. These seven families have been characterized as the only families of digraphs with exactly three complementarity eigenvalues [1], and therefore our results have consequences in this context, showing which families can be determined by the complementarity spectrum. As a particular case, we prove that the θ-digraphs can be characterized by the spectral radius, extending some recent results on this family [2].

引用次数: 0

Additive maps preserving rank-bounded sets of matrices

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-17 DOI: 10.1016/j.laa.2025.01.018

E. Akhmedova , A. Guterman , I. Spiridonov

Let

2 \leq k \leq n

be integers and

{Mat}_{n} (F)

be the linear space of

n \times n

matrices over a field

F

of characteristic different from 2. Denote by

Γ^{\geq k}

the set of matrices in

{Mat}_{n} (F)

of rank greater than or equal to k. The main goal of the present paper is to obtain a characterization of additive maps

f : {Mat}_{n} (F) \to {Mat}_{n} (F)

satisfying

f (Γ^{\geq k}) = Γ^{\geq k}

with either

n < 2 k - 2

F

has characteristic

char (F) = 0

char (F) \geq k

引用次数: 0

A cone-preserving solution to a nonsymmetric Riccati equation

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-17 DOI: 10.1016/j.laa.2025.01.020

Emil Vladu, Anders Rantzer

In this paper, we provide the following simple equivalent condition for a nonsymmetric Algebraic Riccati Equation to admit a stabilizing cone-preserving solution: an associated coefficient matrix must be stable. The result holds under the assumption that said matrix be cross-positive on a proper cone, and it both extends and completes a corresponding sufficient condition for nonnegative matrices in the literature. Further, key to showing the above is the following result which we also provide: in order for a monotonically increasing sequence of cone-preserving matrices to converge, it is sufficient to be bounded above in a single vectorial direction.

引用次数: 0

On symmetric hollow integer matrices with eigenvalues bounded from below

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-17 DOI: 10.1016/j.laa.2025.01.021

Zilin Jiang (姜子麟)

A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define

λ^{⁎} = ρ^{1 / 2} + ρ^{- 1 / 2} \approx 2.01980

, where ρ is the unique real root of

x^{3} = x + 1

. We show that for every

λ < λ^{⁎}

, there exists

n \in N

such that if a symmetric hollow integer matrix has an eigenvalue less than −λ, then one of its principal submatrices of order at most n does as well. However, the same conclusion does not hold for any

λ \geq λ^{⁎}

引用次数: 0

The Nevanlinna formula for matrix Nevanlinna-Pick interpolation

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-17 DOI: 10.1016/j.laa.2025.01.007

Yury Dyukarev

In this paper, we study the matrix Nevanlinna-Pick interpolation problem in the completely indeterminate case. We obtain an explicit formula for the resolvent matrix in terms of rational matrix functions of the first and second kind. Additionally, we describe the set of all solutions to the matrix Nevanlinna-Pick interpolation problem using linear fractional transformations applied to Nevanlinna pairs. This result can be viewed as an analogue of the Nevanlinna formula for the matrix Hamburger moment problem.

引用次数: 0

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