逆柯西矩阵的元之和
环与代数
2023-05-09 v1
摘要
设 x 1 , x 2 , … , x n x_{1},x_{2},\ldots,x_{n} x 1 , x 2 , … , x n n n n y 1 , y 2 , … , y n y_{1},y_{2},\ldots,y_{n} y 1 , y 2 , … , y n n n n n 2 n^{2} n 2 x i + y j x_{i}+y_{j} x i + y j n × n n\times n n × n C : = ( 1 x i + y j ) 1 ≤ i ≤ n , 1 ≤ j ≤ n = ( 1 x 1 + y 1 1 x 1 + y 2 ⋯ 1 x 1 + y n 1 x 2 + y 1 1 x 2 + y 2 ⋯ 1 x 2 + y n ⋮ ⋮ ⋱ ⋮ 1 x n + y 1 1 x n + y 2 ⋯ 1 x n + y n ) . C:=\left( \dfrac{1}{x_{i}+y_{j}}\right) _{1\leq i\leq n,\ 1\leq j\leq n} = \begin{pmatrix} \dfrac{1}{x_{1}+y_{1}} & \dfrac{1}{x_{1}+y_{2}} & \cdots & \dfrac{1}{x_{1}+y_{n}}\\ \dfrac{1}{x_{2}+y_{1}} & \dfrac{1}{x_{2}+y_{2}} & \cdots & \dfrac{1}{x_{2}+y_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \dfrac{1}{x_{n}+y_{1}} & \dfrac{1}{x_{n}+y_{2}} & \cdots & \dfrac{1}{x_{n}+y_{n}} \end{pmatrix}. C := ( x i + y j 1 ) 1 ≤ i ≤ n , 1 ≤ j ≤ n = x 1 + y 1 1 x 2 + y 1 1 ⋮ x n + y 1 1 x 1 + y 2 1 x 2 + y 2 1 ⋮ x n + y 2 1 ⋯ ⋯ ⋱ ⋯ x 1 + y n 1 x 2 + y n 1 ⋮ x n + y n 1 . C C C C C C C − 1 C^{-1} C − 1 ∑ k = 1 n x k + ∑ k = 1 n y k \sum_{k=1}^{n}x_{k}+\sum_{k=1}^{n}y_{k} ∑ k = 1 n x k + ∑ k = 1 n y k 1 x i + y j \dfrac{1}{x_i+y_j} x i + y j 1 min { x i , y j } \min\left\{ x_{i},y_{j}\right\} min { x i , y j }
引用
@article{arxiv.2301.09777,
title = {The entry sum of the inverse Cauchy matrix},
author = {Darij Grinberg},
journal= {arXiv preprint arXiv:2301.09777},
year = {2023}
}
备注
7 pages. Expository note