并非所有 GKK $τ$-矩阵都是稳定的

Diagonal reduction of matrices over Bezout rings of stable range 1 with the Kazimirsky condition

We constuct the theory of diagonalizability for matrices over Bezout rings of stable range 1 with the Kazimirsky condition. It is shown that a ring of stable range 1 with the right (left) Kazimirsky condition is an elementary divisor ring if and only if it is a duo ring.

Reduction of matrices over simple Ore domains

We study the theory of diagonal reductions of matrices over simple Ore domains of finite stable range. We cover the cases of 2-simple rings of stable range 1, Ore domains and certain cases of Bezout domains.

Stable noncommutative polynomials and their determinantal representations

A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of strongly stable linear matrix pencils, i.e., pencils of the form $H+iP_0+P_1x_1+\cdots+P_dx_d$, where $H$ is hermitian and $P_j$ are positive semidefinite matrices. Namely, a noncommutative polynomial is stable if and only if it admits a determinantal representation with a strongly stable pencil. More generally, structure certificates for noncommutative stability are given for linear matrix …

中心化子矩阵代数的导出等价与稳定等价

Derived and stable equivalences of centralizer matrix algebras

该研究通过矩阵初等因子定义方阵新等价关系，将主中心化子矩阵代数的各类范畴等价归约为线性代数问题，证明了置换矩阵对应代数的Morita等价与导出等价等价等结论。

稳定等价与同调维数

Stable equivalences and homological dimensions

本文研究了中心化矩阵代数之间的稳定等价，并证明了相关同调维数的保持以及Alperin--Auslander/Auslander--Reiten猜想的成立。

An infinite analogue of rings with stable rank one

Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings), and include examples like the ring of endomorphisms of a vector space over a field F, and the ring of all row- and column- finite matrices over F. We show that the category of QB-rings is stable under the formation of corners, ideals and quotients, as well as matrices and direct limits. We also give necessary and sufficient conditions for an extension of QB-rings t…

环上矩阵的算术

Arithmetic of matrices over rings

本书研究交换有限生成主理想整环上矩阵环的算术，构建矩阵分解理论，并阐明其与完全线性群子群性质及单侧等价标准形的关系。

Polynomial invariants on matrices and partition, Brauer algebra

We identify the dimension of the centralizer of the symmetric group $\mathfrak{S}_d$ in the partition algebra $\mathcal{A}_d(δ)$ and in the Brauer algebra $\mathcal{B}_d(δ)$ with the number of multidigraphs with $d$ arrows and the number of disjoint union of directed cycles with $d$ arrows, respectively. Using Schur-Weyl duality as a fundamental theory, we conclude that each centralizer is related with the $G$-invariant space $P^d(M_n(\mathbf{k}))^G$ of degree $d$ homogeneous polynomials on $n \times n$ matrices, where $G$ is the orthogonal group and the group of permutation matrices, respecti…