Linear Algebra and its Applications最新文献

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On marginal growth rates of matrix products

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-15 DOI: 10.1016/j.laa.2025.01.013

Jonah Varney , Ian D. Morris

In this article we consider the maximum possible growth rate of sequences of long products of

d \times d

matrices all of which are drawn from some specified compact set which has been normalised so as to have joint spectral radius equal to 1. We define the marginal instability rate sequence associated to such a set to be the sequence of real numbers whose

n^{t h}

entry is the norm of the largest product of length n, and study the general properties of sequences of this form. We describe how new marginal instability rate sequences can be constructed from old ones, extend an earlier example of Protasov and Jungers to obtain marginal instability rate sequences whose limit superior rate of growth matches various non-integer powers of n, and investigate the relationship between marginal instability rate sequences arising from finite sets of matrices and those arising from sets of matrices with cardinality 2. We also give the first example of a finite set whose marginal instability rate sequence is asymptotically similar to a polynomial with non-integer exponent. Previous examples had this property only along a subsequence.

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引用次数: 0

The height of an infinite parallelotope is infinite

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-13 DOI: 10.1016/j.laa.2025.01.011

Alexandr V. Kosyak

We show that if no non-trivial linear combinations of independent vectors

f_{0}, f_{1}, \dots, f_{m} \in R^{\infty}

belongs to

ℓ_{2}

, then all the heights of an infinite parallelotope constructed on vectors

f_{0}, f_{1}, \dots, f_{m}

are infinite. This result is essential in the proof of the irreducibility of unitary representations of some infinite-dimensional groups.

引用次数: 0

When is every linear transformation a sum of a q-potent one and a locally nilpotent one?

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-13 DOI: 10.1016/j.laa.2025.01.012

A.N. Abyzov, D.T. Tapkin

We prove that for each vector space V over

F_{q}

, every linear transformation of V is a sum of a q-potent linear transformation and a locally nilpotent linear transformation.

引用次数: 0

Corrigendum to “A matricial view of the Collatz conjecture” [Linear Algebra Appl. 695 (2024) 163–167]

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-13 DOI: 10.1016/j.laa.2024.12.019

Pietro Paparella

There is a mistake in the proof of Theorem 3 of “A matricial view of the Collatz conjecture” [1] that can not be rectified. As such, a revised statement and proof of Theorem 3 is presented.

引用次数: 0

Darboux transformations and the algebra D(W)

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-10 DOI: 10.1016/j.laa.2025.01.002

Ignacio Bono Parisi, Ines Pacharoni

The problem of finding weight matrices

W (x)

of size

N \times N

such that the associated sequence of matrix-valued orthogonal polynomials are eigenfunctions of a second-order matrix differential operator is known as the Matrix Bochner Problem, and it is closely related to Darboux transformations of some differential operators.

This paper aims to study Darboux transformations between weight matrices and to establish a direct connection with the structure of the algebra

D (W)

of all differential operators that have a sequence of matrix-valued orthogonal polynomials with respect to W as eigenfunctions.

引用次数: 0

Mixed tensor invariants of Lie color algebra

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-10 DOI: 10.1016/j.laa.2025.01.003

Santosha Pattanayak, Preena Samuel

In this paper, we consider the mixed tensor space of a G-graded vector space, where G is a finite abelian group. We obtain a spanning set of invariants of the associated symmetric algebra under the action of a color analogue of the general linear group which we refer to as the general linear color group. As a consequence, we obtain a generating set for the polynomial invariants, under the simultaneous action of the general linear color group, on color analogues of several copies of matrices. We show that in this special case, this is the set of trace monomials, which coincides with the set of generators given by Berele in [2].

引用次数: 0

Spectral methods for matrix product factorization

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-10 DOI: 10.1016/j.laa.2025.01.005

Saieed Akbari , Yi-Zheng Fan , Fu-Tao Hu , Babak Miraftab , Yi Wang

A graph G is factored into graphs H and K via a matrix product if there exist adjacency matrices A, B, and C of G, H, and K, respectively, such that

A = B C

. In this paper, we study the spectral aspects of the matrix product of graphs, including regularity, bipartiteness, and connectivity. We show that if a graph G is factored into a connected graph H and a graph K with no isolated vertices, then certain properties hold. If H is non-bipartite, then G is connected. If H is bipartite and G is not connected, then K is a regular bipartite graph, and consequently, n is even. Furthermore, we show that trees are not factorizable, which answers a question posed by Maghsoudi et al.

引用次数: 0

Refined inertias of nonnegative patterns with positive off-diagonal entries

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-10 DOI: 10.1016/j.laa.2025.01.008

Adam H. Berliner , Minerva Catral , D.D. Olesky , P. van den Driessche

For a positive

n \times n

pattern

A

, it is known that the refined inertia of

A

ri (A)

, is the set of all nonnegative integral 4-tuples

(n_{+}, n_{-}, n_{z}, 2 n_{p})

with

n_{+} + n_{-} + n_{z} + 2 n_{p} = n

and

n_{+} \geq 1

; whereas if

A

has all off-diagonal entries positive but all diagonal entries 0, then

ri (A)

has the additional restriction that

n_{-} \geq 2

. We focus on the intermediate nonnegative patterns, that is those patterns with all off-diagonal entries positive,

k \in {1, \dots, n - 1}

diagonal entries positive and the remaining

n - k

diagonal entries 0. We show that for

k \geq 2

, there is no restriction on

n_{-}

for the refined inertia set, but

n_{-} \geq 1

for

k = 1

. We do this by constructing nonnegative matrix realizations for the patterns with

k = 1

and 2 using the centralizer method, matrix bordering and superpattern results.

引用次数: 0

Quasi-isometric liftings for operators similar to contractions

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-10 DOI: 10.1016/j.laa.2025.01.006

Laurian Suciu, Andra-Maria Stoica

A class of quasi-isometric liftings for the operators T similar to contractions in Hilbert spaces

H

is studied. These liftings are isometric operators on their ranges, and are naturally induced by T and an invertible intertwiner of T with a contraction. In the case when T is a quasicontraction, meaning that T is contractive on its range, we obtain a quasi-isometric lifting on a space

K \supset H

, which is isometric on

K ⊖ H

. Some liftings with closed ranges, or even similar to quasinormal partial isometries are mentioned. Additionally, we study the isomorphic minimal quasi-isometric liftings for T, as well as the uniqueness property of such liftings. Our results show similarities with those from the isometric dilation theory for contractions, although our context is more general than that of the latter.

引用次数: 0

The triangulant

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2025-01-10 DOI: 10.1016/j.laa.2025.01.004

Tamás Bencze , Péter E. Frenkel

We introduce the triangulant of two matrices, and relate it to the existence of orthogonal eigenvectors. We also use it for a new characterization of mutually unbiased bases. Generalizing the notion, we introduce higher order triangulants of two matrices, and relate them to the existence of nontrivially intersecting invariant subspaces of complementary dimensions.

引用次数: 0

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