标题： 广义范德蒙矩阵及相关课题 显示英文标题

标题： Confluent Vandermonde matrix and related topics

摘要： 在本笔记中，我们探讨了连带范德蒙矩阵与若干数学主题之间的联系，包括插值多项式、哈斯导数、LU 分解、伴随矩阵及其若尔当形式，以及中国剩余定理。通过基于选定节点上的多项式评估和导数计算的统一方法，我们提供了易于理解的证明，这些证明不仅阐明了关键结果，而且还为经验丰富的研究人员和该领域的初学者提供了见解。 显示英文摘要

摘要： In this note, we explore the connections between the confluent Vandermonde matrix and several mathematical topics, including interpolating polynomials, Hasse derivatives, LU factorization, companion matrices and their Jordan forms, and the Chinese remainder theorem. Using a unified approach based on polynomial evaluations and derivative computations at selected points, we provide accessible proofs that not only clarify key results but also offer insights for both experienced researchers and those new to the subject.