数值矩阵分解
Numerical Matrix Decomposition
AI总结：
本综述介绍数值线性代数与矩阵分析的概念和数学工具，为后续讲解矩阵分解技术及其机器学习应用做铺垫，因篇幅有限未覆盖全部结论，建议参阅线性代数文献补充相关内容。
摘要：
1954年，Alston S. Householder出版了《数值分析原理》，这是最早的现代矩阵分解研究著作之一，该书推崇（分块）LU分解——即将矩阵分解为下三角矩阵与上三角矩阵的乘积。如今，由于神经网络拟合中反向传播算法的发展，矩阵分解已成为机器学习领域的核心技术。本综述的唯一目标是对数值线性代数与矩阵分析中的概念和数学工具进行自包含的介绍，以便在后续章节中顺畅地引入矩阵分解技术及其应用。不过我们清楚地意识到，自己无法涵盖所有与矩阵分解相关的实用且有价值的结论，并且可供展开讨论的篇幅有限，例如无法对欧几里得空间、埃尔米特空间、希尔伯特空间以及复域中的相关内容分别展开分析。我们建议读者参阅线性代数领域的文献，以获取相关领域更详细的介绍。关键词：矩阵分解的存在性与计算、浮点运算（flops）、低秩近似、主元、LU/PLU分解、CR/CUR/Skeleton分解、坐标变换、ULV/URV分解、秩分解、秩揭示分解、更新/降阶更新、张量分解。
英文摘要：
In 1954, Alston S. Householder published \textit{Principles of Numerical Analysis}, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower and upper triangular matrices. And now, matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in numerical linear algebra and matrix analysis in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of the Euclidean space, Hermitian space, Hilbert space, and things in the complex domain. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields. Keywords: Existence and computing of matrix decompositions, Floating point operations (flops), Low-rank approximation, Pivot, LU/PLU decomposition, CR/CUR/Skeleton decomposition, Coordinate transformation, ULV/URV decomposition, Rank decomposition, Rank revealing decomposition, Update/downdate, Tensor decomposition.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
