论文标题
在Cartan分解上的经典随机矩阵合奏
On the Cartan Decomposition for Classical Random Matrix Ensembles
论文作者
论文摘要
我们通过巩固对称空间与经典随机矩阵合奏之间的联系来完成戴森的梦想。先前的工作集中在对称空间与许多但不是所有经典随机矩阵集合之间的一对一对应关系上。这项工作表明，我们可以通过使用替代坐标系统从Cartan的对称空间完全捕获所有经典的随机矩阵合奏。最后，我们必须放开一对一信件的概念。 我们强调，传统上，数学家对KAK分解只是一个在对称空间上的坐标系统，尽管它是一个美丽的空间。但是，其他矩阵因子化，尤其是来自数值线性代数的广义奇异值分解表明自己是完全有效的坐标系，表明一个对称空间可以导致许多经典的随机矩阵理论。 我们在此数值线性代数观点与广义cartan分解理论之间建立了联系。反过来，这使我们能够从单个对称空间中产生更多随机矩阵理论。再次，这些随机矩阵理论是由矩阵因素化引起的，我们不知道在文献中出现的矩阵理论。
We complete Dyson's dream by cementing the links between symmetric spaces and classical random matrix ensembles. Previous work has focused on a one-to-one correspondence between symmetric spaces and many but not all of the classical random matrix ensembles. This work shows that we can completely capture all of the classical random matrix ensembles from Cartan's symmetric spaces through the use of alternative coordinate systems. In the end, we have to let go of the notion of a one-to-one correspondence. We emphasize that the KAK decomposition traditionally favored by mathematicians is merely one coordinate system on the symmetric space, albeit a beautiful one. However, other matrix factorizations, especially the generalized singular value decomposition from numerical linear algebra reveal themselves to be perfectly valid coordinate systems revealing that one symmetric space can lead to many classical random matrix theories. We establish the connection between this numerical linear algebra viewpoint and the theory of generalized Cartan decomposition. This in turn allows us to produce yet more random matrix theories from a single symmetric space. Yet again these random matrix theories arise from matrix factorizations, through ones that we are not aware have appeared in the literature.
