五十三个矩阵分解：一个系统的方法
Fifty Three Matrix Factorizations: A systematic approach
摘要：
诸如奇异值分解（SVD）之类的矩阵分解的成功激发了对更多分解的搜索。我们列出了53种矩阵分解，其中大部分我们认为是新的。我们的系统方法，启发广义Cartan分解的李理论，也包括已知的分解，如奇异值分解，对称特征分解，CS分解，双曲奇异值分解，结构奇异值分解，高木分解，和其他从而涵盖熟悉的矩阵分解，以及那些等待被发现。我们认为，李理论已经以这样或那样的方式潜伏在非常成功的矩阵计算领域的基础上，并在许多计算领域中经常使用。本文研究了矩阵分解的Cartan分解和鲜为人知的广义Cartan分解的结果。我们相信，这些因式分解一旦正确识别，可以导致进一步的工作算法计算和应用。
英文摘要：
The success of matrix factorizations such as the singular value decomposition (SVD) has motivated the search for even more factorizations. We catalog 53 matrix factorizations, most of which we believe to be new. Our systematic approach, inspired by the generalized Cartan decomposition of Lie theory, also encompasses known factorizations such as the SVD, the symmetric eigendecomposition, the CS decomposition, the hyperbolic SVD, structured SVDs, the Takagi factorization, and others thereby covering familiar matrix factorizations as well as ones that were waiting to be discovered. We suggest that Lie theory has one way or another been lurking hidden in the foundations of the very successful field of matrix computations with applications routinely used in so many areas of computation. In this paper, we investigate consequences of the Cartan decomposition and the little known generalized Cartan decomposition for matrix factorizations. We believe that these factorizations once properly identified can lead to further work on algorithmic computations and applications.
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