关于矩阵的低秩扰动

秩1矩阵的张量积的线性秩变换

Linear rank preservers of tensor products of rank one matrices

令$n_1,\ldots,n_k $为大于或等于2的整数。本文给出了线性映射$ϕ: M_{n_1\cdots n_k}\rightarrow M_{n_1\cdots n_k}$的特征，使得$${\mathrm rank}\,(ϕ(A_1\otimes \cdots \otimes A_k))=1\quad\hbox{whenever}\quad{\mathrm rank}\, (A_1\otimes \cdots \otimes A_k)=1 \quad \hbox{for all}\quad A_i \in M_{n_i},\, i = 1,\dots,k.$$应用这个结果，我们推广了最近关于一类特殊矩阵保秩的线性映射的两个结果.

非结构矩阵秩一扰动的特征值

Eigenvalues of rank one perturbations of unstructured matrices

令$A$是固定的复矩阵，并且令$u,v$是两个向量。矩阵$A+τuv^\top $ $(τ\in\mathbb{R})$的特征值形成相交曲线的系统。研究了交点对矢量$u,v$的依赖性。

A note on a conjecture concerning rank one perturbations of singular M-matrices

A conjecture from a paper by J. Bierkens and A.C.M. Ran concerning the location of eigenvalues of rank one perturbations of singular M-matrices is shown to be false in dimension four and higher, but true for dimension two, as well as for dimension three with an additional condition on the perturbation.

Almost commuting matrices with respect to the rank metric

We show that if A_1, A_2, ... , A_n are square matrices, each of them is either unitary or self-adjoint, and they almost commute with respect to the rank metric, then one can find commuting matrices B_1, B_2, ... , B_n that are close to the matrices A_i in the rank metric.

关于Helson矩阵：矩量问题、非负性、有界性和有限秩

On Helson matrices: moment problems, non-negativity, boundedness, and finite rank

我们研究Helson矩阵（也称为乘法Hankel矩阵），即$M(α) = \{α(nm)\}_{n,m=1}^\infty$形式的无限矩阵，其中$α$是复数序列。Helson矩阵被认为是$\ell^2(\mathbb{N})$上的线性算子。通过将Helson矩阵解释为可数多变量的Hankel矩阵，我们使用多元矩问题的理论来证明$M(α)$是非负的当且仅当$α$是$\mathbb{R}^\infty$上的测度$μ$的矩序列，假设$α$不会增长得太快。然后我们刻画了非负有界Helson矩阵$M(α)$为可数多变量Hardy空间的相应矩测度$μ$为Carleson测度的矩阵.最后，我们给出了有限秩Helson矩阵的完整描述，与Hankel矩阵的经典Kronecker定理并行。