Factorization of Symmetric Matrices and Actions on Σ_k

Cubic forms having matrix factorizations by Hessian matrices

Using a part of XJC-correspondence by Pirio and Russo, we classify cubic forms $f$ whose Hessian matrices induce matrix factorizations of themselves. When it defines a reduced hypersurface, it satisfies the "secant-singularity" correspondence, that is, it coincides with the secant locus of its singular locus. In particular, when $f$ is irreducible, its singular locus is either one of four Severi varieties.

4×4矩阵Hadamard积分解的计算求解

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。

结构化矩阵分解长度

Structured matrix factorization length

研究将任意矩阵分解为具有特定结构（如Toeplitz、Hankel等）的矩阵乘积所需的最小个数，引入结构化矩阵分解长度及其边界概念，并计算各类结构的分解簇维数。

Normal Factorization in $SL(2,Z)$ and the Confluence of Singular Fibers in Elliptic Fibrations

In this article we obtain a result about the uniqueness of factorization in terms of conjugates of the matrix $U=(\xymatrix{1 & 1 0 & 1})$, of some matrices representing the conjugacy classes of those elements of $SL(2,Z)$ arising as the monodromy around a singular fiber in an elliptic fibration (i.e. those matrices that appear in Kodaira's list).

Integral points on varieties defined by matrix factorization into elementary matrices

Let ${\mathcal O}$ be the ring of $S$-integers in a number field $K$. For $A\in\rm{SL}_{2}(\mathcal{O})$ and $k\geq 1$, we define matrix-factorization varieties $V_k(A)$ over ${\mathcal O}$ which parametrize factoring $A$ into a product of $k$ elementary matrices; the equations defining $V_k(A)$ are written in terms of Euler's continuant polynomials. We show that the $V_k(A)$ are rational $(k-3)$-folds with an inductive fibration structure. We combine this geometric structure with arithmetic results to study the Zariski closure of the ${\mathcal O}$-points of $V_k(A)$. We prove that for $k\geq…