Linear Algebra and its Applications最新文献

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A note on a conjecture from distillability of quantum entanglement 关于量子纠缠可蒸馏性猜想的说明

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-11-19 DOI: 10.1016/j.laa.2024.11.012

Weng Kun Sio, Che-Man Cheng

A conjecture from the distillability of quantum entanglement is that when A and B are

4 \times 4

trace zero complex matrices and

{‖ A ‖}^{2} + {‖ B ‖}^{2} = 1 / 4

(where

‖ \cdot ‖

is the Frobenius norm), the sum of squares of the largest two singular values of

A \otimes I_{4} + I_{4} \otimes B

does not exceed 1/2. In this paper, the conjecture is proved when

(i)
A or B is unitarily similar to a direct sum of $2 \times 2$ trace zero matrices;
(ii)
A and B are unitarily similar to matrices, when partitioned into $2 \times 2$ blocks, having zero diagonal blocks.

量子纠缠的可提炼性的一个猜想是：当 A 和 B 是 4×4 痕量为零的复矩阵且‖A‖2+‖B‖2=1/4（其中‖⋅‖是弗罗贝尼斯规范）时，A⊗I4+I4⊗B 的最大两个奇异值的平方和不超过 1/2 。本文证明了以下猜想：(i) A 或 B 与 2×2 痕零矩阵的直接和具有单位相似性；(ii) A 和 B 被分割成 2×2 块时与矩阵具有单位相似性，且对角块为零。

引用次数: 0

Tight frames generated by a graph short-time Fourier transform 由图形短时傅立叶变换生成的紧帧

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-11-19 DOI: 10.1016/j.laa.2024.11.014

Martin Buck, Kasso A. Okoudjou

A graph short-time Fourier transform is defined using the eigenvectors of the graph Laplacian and a graph heat kernel as a window parametrized by a nonnegative time parameter t. We show that the corresponding Gabor-like system forms a frame for

C^{d}

and gives a description of the spectrum of the corresponding frame operator in terms of the graph heat kernel and the spectrum of the underlying graph Laplacian. For two classes of algebraic graphs, we prove the frame is tight and independent of the window parameter t.

我们利用图拉普拉卡的特征向量和图热核定义了图短时傅立叶变换，并将其作为由非负时间参数 t 参数化的窗口。我们证明了相应的类 Gabor 系统形成了 Cd 的框架，并用图热核和底层图拉普拉卡的频谱描述了相应框架算子的频谱。对于两类代数图，我们证明了框架是紧密的，且与窗口参数 t 无关。

引用次数: 0

Upper bounds for the rank of powers of quadrics 四面体幂级数的上限

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-11-19 DOI: 10.1016/j.laa.2024.11.009

Cosimo Flavi

We establish an upper bound for the rank of every power of an arbitrary quadratic form. Specifically, for any

s \in N

, we prove that the s-th power of a quadratic form of rank n grows as

n^{s}

. Furthermore, we demonstrate that its rank is subgeneric for all

n > {(2 s - 1)}^{2}

我们为任意二次函数形式的每个幂的秩建立了一个上限。具体来说，对于任意 s∈N，我们证明秩为 n 的二次函数形式的 s 次幂随 ns 增长。此外，我们还证明了对于所有 n>(2s-1)2，它的秩都是子代的。

引用次数: 0

Irreducible matrix representations of quaternions 四元数的不可减矩阵表示

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-11-17 DOI: 10.1016/j.laa.2024.11.010

Yu Chen

We determine all irreducible real and complex matrix representations of quaternions and classify them up to equivalence. More over, we show that there is a one-to-one correspondence between the equivalence classes of the irreducible matrix representations and those of the field homomorphisms from the real numbers to the complex numbers.

我们确定了四元数的所有不可还原实数和复数矩阵表示，并对它们进行了等价分类。此外，我们还证明了不可还原矩阵表示的等价类与从实数到复数的场同构类之间存在一一对应关系。

引用次数: 0

Jordan embeddings and linear rank preservers of structural matrix algebras 结构矩阵代数的约旦嵌入和线性秩保护器

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-11-16 DOI: 10.1016/j.laa.2024.11.013

Ilja Gogić, Mateo Tomašević

We consider subalgebras

A

of the algebra

M_{n}

n \times n

complex matrices that contain all diagonal matrices, known in the literature as the structural matrix algebras (SMAs).

Let

A \subseteq M_{n}

be an arbitrary SMA. We first show that any commuting family of diagonalizable matrices in

A

can be intrinsically simultaneously diagonalized (i.e. the corresponding similarity can be chosen from

A

). Using this, we then characterize when one SMA Jordan-embeds into another and in that case we describe the form of such Jordan embeddings. As a consequence, we obtain a description of Jordan automorphisms of SMAs, generalizing Coelho's result on their algebra automorphisms. Next, motivated by the results of Marcus-Moyls and Molnar-Šemrl, connecting the linear rank-one preservers with Jordan embeddings

M_{n} \to M_{n}

and

T_{n} \to M_{n}

(where

T_{n}

is the algebra of

n \times n

upper-triangular matrices) respectively, we show that any linear unital rank-one preserver

A \to M_{n}

is necessarily a Jordan embedding. As the converse fails in general, we also provide a necessary and sufficient condition for when it does hold true. Finally, we obtain a complete description of linear rank preservers

A \to M_{n}

, as maps of the form

X \mapsto S (P X + (I - P) X^{t}) T

, for some invertible matrices

S, T \in M_{n}

and a central idempotent

P \in A

我们考虑的是包含所有对角矩阵的 n×n 复矩阵代数 Mn 的子代数 A，文献称之为结构矩阵代数（SMA）。我们首先证明，A 中任何可对角化矩阵的换向族都可以内在地同时对角化（即可以从 A 中选择相应的相似性）。利用这一点，我们可以描述一个 SMA 乔丹嵌入到另一个 SMA 中的情况，并在这种情况下描述这种乔丹嵌入的形式。因此，我们得到了对 SMA 的乔丹自动形的描述，推广了科埃略关于其代数自动形的结果。接下来，受马库斯-莫伊尔斯和莫尔纳-舍姆尔的结果的启发，我们分别将线性秩一预言器与乔丹嵌入 Mn→Mn 和 Tn→Mn （其中 Tn 是 n×n 上三角矩阵代数）联系起来，证明任何线性单元秩一预言器 A→Mn 必然是一个乔丹嵌入。由于反向一般不成立，我们还提供了一个必要条件和充分条件来说明何时反向成立。最后，对于一些可逆矩阵 S,T∈Mn 和一个中心幂等 P∈A，我们得到了线性秩预言器 A→Mn 的完整描述，即形式为 X↦S(PX+(I-P)Xt)T 的映射。

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引用次数: 0

Matrix diagonalisation in sesquilinear symplectic spaces 倍线性交映空间中的矩阵对角化

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-11-14 DOI: 10.1016/j.laa.2024.11.007

Tanvi Jain , Kirti Kajla

The symplectic inner product on

C^{2 n}

is the sesquilinear form given by

[x, y] = 〈 x, J_{2 n} y 〉,

where

J_{2 n}

is the real skew-symmetric, orthogonal

2 \times 2

block matrix

[\begin{matrix} 0 & I_{n} \\ - I_{n} & 0 \end{matrix}]

. We derive results analogous to the spectral theorem and singular value decomposition for complex matrices such as Hamiltonian and J-normal matrices, in the sesquilinear symplectic inner product spaces.

C2n 上的交映内积是由[x,y]=〈x,J2ny〉给出的倍线性形式，其中 J2n 是实倾斜对称正交 2×2 矩阵 [0In-In0]。在倍线性交映内积空间中，我们推导出类似于哈密顿矩阵和 J 正矩阵等复数矩阵的谱定理和奇异值分解的结果。

引用次数: 0

Eigenvalues of Toeplitz matrices emerging from finite differences for certain ordinary differential operators 某些常微分算子的有限差分产生的托普利兹矩阵特征值

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-11-14 DOI: 10.1016/j.laa.2024.11.008

M. Bogoya , A. Böttcher , S.M. Grudsky

We consider Hermitian Toeplitz matrices emerging from finite linear combinations with non-negative coefficients of the differential operators

{(- 1)}^{k} d^{2 k} / d x^{2 k}

over the interval

(0, 1)

after discretizing them on a uniform grid of step size

1 / (n + 1)

. The collective distribution in the Szegő–Weyl sense of the eigenvalues of these matrices as n goes to infinity can be described by GLT theory. However, we focus on the asymptotic behavior of the individual eigenvalues, on both the inner eigenvalues in the bulk and on the extreme eigenvalues. The difficulty of the problem is that not only the order of the matrices depends on n but also their so-called symbols. Our main results are third order asymptotic formulas for the eigenvalues in the case

k ⩽ 2

. These results reveal some basic phenomena one should expect when considering the problem in full generality.

我们考虑的是在步长为 1/(n+1) 的均匀网格上离散化后，由区间 (0,1) 上微分算子 (-1)kd2k/dx2k 的非负系数有限线性组合产生的赫米蒂托普利兹矩阵。这些矩阵的特征值随着 n 变为无穷大时的 Szegő-Weyl 意义上的集体分布可以用 GLT 理论来描述。不过，我们关注的重点是单个特征值的渐近行为，既包括主体中的内部特征值，也包括极端特征值。问题的难点在于，矩阵的阶数不仅取决于 n，还取决于它们的所谓符号。我们的主要结果是 k⩽2 情况下特征值的三阶渐近公式。这些结果揭示了我们在全面考虑这个问题时应该预料到的一些基本现象。

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引用次数: 0

Computing eigenvalues for products of two classes of sign regular matrices to high relative accuracy 高相对精度计算两类符号正则矩阵乘积的特征值

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-11-13 DOI: 10.1016/j.laa.2024.11.006

Xiaoxiao Ma , Yingqing Xiao , Zhao Yang

In this paper, we consider how to accurately solve the product eigenvalue problem for the class of sign regular (SR) matrices with signature

(1, \dots, 1, - 1)

and the class of totally nonnegative (TN) matrices, which tend to be extremely ill-conditioned. We present algorithms with

O (n^{3})

complexity to accurately compute the parameter matrices of products of TN matrices and SR matrices with signature

(1, \dots, 1, - 1)

. Based on the accurate parameter matrices, all eigenvalues of the product matrix are computed to high relative accuracy. Numerical experiments are provided to confirm the claimed high relative accuracy.

在本文中，我们考虑了如何精确求解签名为 (1,⋯,1,-1) 的符号正则（SR）矩阵和完全非负（TN）矩阵的乘积特征值问题。我们提出了复杂度为 O(n3) 的算法，可以精确计算 TN 矩阵和符号为 (1,...,1,-1) 的 SR 矩阵乘积的参数矩阵。基于精确的参数矩阵，乘积矩阵的所有特征值都能以较高的相对精度计算出来。提供的数值实验证实了所宣称的高相对精度。

引用次数: 0

Comprehensive classification of the algebra generated by two idempotent matrices 由两个幂等矩阵生成的代数的综合分类

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-11-07 DOI: 10.1016/j.laa.2024.11.005

Rounak Biswas, Falguni Roy

For two idempotent matrix

P, Q \in C^{n \times n}

, let alg

(I_{n}, P, Q)

denote the smallest subalgebra of

C^{n \times n}

that contains

P, Q

and the identity matrix

I_{n}

. This paper provides a complete classification of alg

(I_{n}, P, Q)

without imposing any restrictions on P and Q. As a result of this classification, the issue of group invertibility within alg

(I_{n}, P, Q)

is fully resolved.

对于两个幂等矩阵 P,Q∈Cn×n, 让 alg(In,P,Q) 表示 Cn×n 中包含 P,Q 和同一矩阵 In 的最小子代数。本文在不对 P 和 Q 施加任何限制的情况下，对 alg(In,P,Q) 进行了完整的分类。

引用次数: 0

Quantum subspace controllability implying full controllability 量子子空间可控性意味着完全可控性

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2024-11-07 DOI: 10.1016/j.laa.2024.11.002

Francesca Albertini , Domenico D'Alessandro

In the analysis of controllability of finite dimensional quantum systems, subspace controllability refers to the situation where the underlying Hilbert space splits into the direct sum of invariant subspaces, and, on each of such invariant subspaces, it is possible to generate any arbitrary unitary operation using appropriate control functions. This is a typical situation in the presence of symmetries for the dynamics.

We investigate whether and when if subspace controllability is verified, the addition of an extra Hamiltonian to the dynamics implies full controllability of the system. Under the natural (and necessary) condition that the new Hamiltonian connects all the invariant subspaces, we show that this is always the case, except for a very specific case we shall describe. Even in this specific case, a weaker notion of controllability, controllability of the state (Pure State Controllability) is verified.

在有限维量子系统的可控性分析中，子空间可控性指的是底层希尔伯特空间分裂成不变子空间的直接和，在每个不变子空间上，都可以使用适当的控制函数产生任意的单元操作。我们研究了如果子空间可控性得到验证，那么在动力学中增加一个额外的哈密尔顿是否意味着系统的完全可控性。在新哈密顿连接所有不变子空间的自然（必要）条件下，我们证明情况总是如此，除了我们将描述的一种非常特殊的情况。即使在这种特殊情况下，也能验证较弱的可控性概念，即状态可控性（纯状态可控性）。

引用次数: 0

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