In 1954, Alston S. Householder published 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.

翻译：1954年，Alston S. Householder出版了《数值分析原理》，这是最早采用（分块）LU分解——即将矩阵分解为下三角矩阵与上三角矩阵乘积的现代矩阵分解论著之一。如今，矩阵分解已成为机器学习的核心技术，这在很大程度上归功于反向传播算法在神经网络拟合中的发展。本综述的唯一目的是对数值线性代数和矩阵分析中的概念与数学工具进行自成体系的介绍，以便在后继章节中无缝引入矩阵分解技术及其应用。然而，我们清楚地认识到自身无法涵盖所有关于矩阵分解的有用且有趣的结果，同时受限于讨论范围的局限性，例如未能分别分析欧几里得空间、埃尔米特空间、希尔伯特空间及复数域中的相关内容。我们建议读者参阅线性代数领域的文献以获取相关领域的更详细介绍。