H. Chen, H. Kose, Y. Kurtulmaz
An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of matrices over projective-free rings. These extend many known results on strongly clean matrices over commutative local rings.
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We investigate projective covers of cyclically presented modules, characterizing the rings over which every cyclically presented module has a projective cover as the rings $R$ that are Von Neumann regular modulo their Jacobson radical $J(R)$ and in which idempotents can be lifted modulo $J(R)$. Cyclically presented modules naturally appear in the study of factorizations of elements in non-necessarily commutative integral domains. One of the possible applications is to the modules $M_R$ whose endomorphism ring $E:=(M_R)$ is Von Neumann regular modulo $J(E)$ and in which idempotents lift modulo …
An element $a$ in a ring $R$ is strongly J-clean if it is the sum of an idempotent and an element in the Jacobson radical that commutes. We characterize the strongly J-clean $2\times 2$ matrices over 2-projective-free non-commutative rings.
The $ϕ$-conjugation of quaternionic matrices and generalized Autonne-Takagi factorization
本文针对模为1的四元数$\phi$，研究四元数矩阵的$\phi$-共轭相关性质，并将Autonne-Takagi分解推广至所有单位四元数，肯定回答了Horn与Zhang提出的问题。
Idempotent factorization on some matrices over quadratic integer rings
本文针对Cossu和Zanardo关于二次整数环上特定奇异矩阵幂等分解的猜想，构造了支持与反对该猜想的矩阵类，并证明存在不能分解为两个幂等矩阵乘积的此类矩阵。
Given an arbitrary monic polynomial $f$ over a field $F$ of characteristic 0, we use companion matrices to construct a polynomial $M_f\in F[X]$ of minimum degree such that for each root $α$ of $f$ in the algebraic closure of $F$, $M_f(α)$ is equal to the multiplicity $m(α)$ of $α$ as a root of $f$. As an application of $M_f$ we give a new method to compute in $F[X]$ each component of the square-free factorization $f=P_1P_2^2\cdots P_m^m$, where $P_k$ is the product of all $X-α$ with $m(α)=k$, for $k=1, \dots, m=\max m(α)$.
N-fold module factorizations: triangle equivalences and recollements
本文引入推广交换与非交换分解的$n$重模分解，通过确定合适子范畴提供了Chen定理三角等价的$n$-analogue，并研究了涉及高重模分解稳定范畴的recollements。
We study exponential factorization of invertible matrices over unital complex Banach algebras. In particular, we prove that every invertible matrix with entries in the algebra of holomorphic functions on a closed bordered Riemann surface can be written as a product of two exponents of matrices over this algebra. Our result extends similar results proved earlier in [KS] and [L] for $2\times 2$ matrices.
Let $T_n(q)$ be the ring of lower triangular matrices of order $n \geq 2$ with entries from the finite field $F(q)$ of order $q \geq 2$ and let ${^2T_n(q)}$ denote its free left module. For $n=2,3$ it is shown that the projective line over $T_n(q)$ gives rise to a set of $(q+1)^{(n-1)}q^{\frac{3(n-1)(n-2)}{2}}$ affine planes of order $q$. The points of such an affine plane are non-free cyclic submodules of ${^2T_n(q)}$ not contained in any non-unimodular free cyclic submodule of ${^2T_n(q)}$ and its lines are points of the projective line. Furthermore, it is demonstrated that each affine plane…
It is well known that any nonsingular M-matrix admits an LU factorization into M-matrices (with L and U lower and upper triangular respectively) and any singular M-matrix is permutation similar to an M-matrix which admits an LU factorization into M-matrices. Varga and Cai establish necessary and sufficient conditions for a singular M-matrix (without permutation) to allow an LU factorization with L nonsingular. We generalize these results in two directions. First, we find necessary and sufficient conditions for the existence of an LU factorization of a singular M-matrix where L and U are both p…
In this note, firstly we give an easy proof of the factorization of symmetric matrices (see [Mos] math-ph/0203023), then we use it to prove the well-known fact that the automorphism group of a non-degenerate symmetric bilinear form acts transitively on the locus of isotropic subspaces Σ_k.
An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element is the product of irreducibles, and it satisfies the ACCP if every ascending chain of principal ideals eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies the finite factorization property (FFP) if every element has only finitely many factorizations, and it satisfies the bounded factorization property (BFP) if for each element there is a common bound for the number of atoms in each of its factorizations.…
A matrix is $k$-nonnegative if all its minors of size $k$ or less are nonnegative. We give a parametrized set of generators and relations for the semigroup of $k$-nonnegative $n\times n$ invertible matrices in two special cases: when $k = n-1$ and when $k = n-2$, restricted to unitriangular matrices. For these two cases, we prove that the set of $k$-nonnegative matrices can be partitioned into cells based on their factorizations into generators, generalizing the notion of Bruhat cells from totally nonnegative matrices. Like Bruhat cells, these cells are homeomorphic to open balls and have a to…
Module factorizations
本文研究任意环中正则正规元素的模分解范畴，通过余核函子建立与Gorenstein投射模的联系，并推广了Eisenbud矩阵分解定理。
Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices
本文通过下三角、三对角和上三角矩阵分解，给出两个单对矩阵之和及其逆的半闭式求逆公式，并应用于特定 Gram 矩阵符号逆推导与数值稳定性分析。
We present a simple proof of the factorization of (complex) symmetric matrices into a product of a square matrix and its transpose, and discuss its application in establishing a uniqueness property of certain antilinear operators.
Any set of $σ$-Hermitian matrices of size $n \times n$ over a field with involution $σ$ gives rise to a projective line in the sense of ring geometry and a projective space in the sense of matrix geometry. It is shown that the two concepts are based upon the same set of points, up to some notational differences.
Idempotent factorizations of singular $2\times 2$ matrices over quadratic integer rings
该研究聚焦二次整数环上奇异2×2矩阵的幂等因子分解问题，利用$SL_2(D)$由初等矩阵生成的结论，证明d>0时含零行/零列的矩阵及列-行矩阵均存在幂等分解，补充了d<0时存在不可分解矩阵的相关结论。
Idempotent factorization of matrices over a Prüfer domain of rational functions
本文针对Prüfer非Bézout域的公开问题，研究了有理函数极小Dress环（一种Prüfer域上2×2奇异矩阵的幂等分解问题，给出了保证其可分解为幂等矩阵乘积的若干条件。
In this paper we characterize EP operators through the existence of different types of factorizations. Our results extend to EP operators existing characterizations for EP matrices and give new characterizations both for EP matrices and EP operators.
Factorizations into Normal Matrices in Indefinite Inner Product Spaces
研究显示任何非奇异矩阵可分解为三个正常矩阵乘积，其中一个是酉矩阵，另一个是具有右半平面特征值的自伴矩阵，第三个是中性正常反置矩阵。
Factorizations into idempotent factors of matrices over Prüfer domains
针对刻画满足幂等矩阵分解性质的整环这一经典问题，研究证明满足(ID2)的整环必为满足特定可逆矩阵分解条件的Prüfer域，并验证了一大类平面曲线坐标环与Int(Z)符合相关猜想的等价形式。
A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A=BB^T. If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A. In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3. Failure of this algorithm implies that A does not have cp-rank 3. Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that vanish at the boundary of an interval and are orthonormal with respect to a certain inner p…
Projective Banach Lie bialgebras, the projective Yang-Baxter equation and projective Poisson-Lie groups
本文引入射影Banach李双代数、射影Yang-Baxter方程及射影Poisson-Lie群，证明拟三角射影r-矩阵产生射影Banach李双代数，射影Poisson-Lie群微分具有该结构，并给出三角射影r-矩阵的等价描述。
Computational Resolution of Hadamard Product Factorization for $4 \times 4$ Matrices
本文通过穷举搜索和机器学习方法研究4×4满秩矩阵的Hadamard积分解问题，发现26.3%的反例，并揭示可表达矩阵位于近似10维簇上，表明存在深层代数约束。
Cholesky decomposition for symmetric matrices over finite fields
本文将一般Cholesky分解扩展至有限域上渐近密度为1的LPM锥，证明其与逐项Frobenius映射兼容，并利用Cholesky结构定义群运算及枚举子锥。
Jordan algebras, hemiplex numbers, and the Cholesky decomposition of arbitrary symmetric matrices
研究任意对称矩阵Cholesky分解，通过类比正定矩阵与特定二次方程，利用半复数这种非结合代数实现分解，无需pivoting保持带状结构，对奇异矩阵可生成零空间参数化，为求解对称线性方程增添实用工具。
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