Factoring Permutation Matrices Into a Product of Tridiagonal Matrices

Factorization of permutations

We consider the problem of factoring permutations as a product of special types of transpositions, namely, those transpositions involving two positions with bounded distances. In particular, we investigate the minimum number, $δ$, such that every permutation can be factored into no more than $δ$ special transpositions. This study is related to sorting algorithms, Cayley graphs, and genomics.

将无模式排列分解为可分离排列

Factoring Pattern-Free Permutations into Separable ones

本文证明任意避免固定模式的排列都可分解为至多常数个可分离排列的乘积，并利用排列宽度给出线性时间分解算法，进而应用于有界 twin-width 图类的一阶转导刻画。

Tridiagonal matrices with nonnegative entries

In this paper we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let $d$ denote a nonnegative integer. Let $A$ denote a matrix in $\matR$ and let $\{þ_i\}_{i=0}^d$ denote the roots of the characteristic polynomial of $A$. We say $A$ is multiplicity-free whenever these roots are mutually distinct and contained in $\R$. In this case $E_i$ will denote the primitive idempotent of $A$ associated with $þ_i$ $(0 \leq i \leq d)$. We say $A$ is symmetrizable whenever the…

使用GEPP的随机置换

Random permutations using GEPP

本文研究随机矩阵在部分主元高斯消去中诱导的置换，分类蝴蝶矩阵的置换结构，分析GOE、GUE等模型的非均匀性及渐近行为，并揭示LAPACK实现细节对主元概率的影响，建立随机矩阵与置换组合学的新联系。