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Cartan's coordinate system (KAK) and GSVD coordinate systems (K 1 AK 2 ) on the Grassmannian manifold O(n)/(O(n − s) × O(s)).
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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...
Citations
... The following papers relate to random matrix theory and eigenvalue statistics. [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] The ...
... It turned out that this extended classification is in on-to-one correspondance with Cartan's old classification of symmetric spaces [3]. This correspondance has motivated the notion that the relationship between random matrices and symmetric spaces extends well beyond symmetric cones, and is of a general nature (for example [4] or [5,6]). ...
... In [4], this is used to recover the Jacobi ensembles of random matrix theory. ...
The integral of a function $f$ defined on a symmetric space $M \simeq G/K$ may be expressed in the form of a determinant (or Pfaffian), when $f$ is $K$-invariant and, in a certain sense, a tensor power of a positive function of a single variable. The paper presents a few examples of this idea and discusses future extensions. Specifically, the examples involve symmetric cones, Grassmann manifolds, and classical domains.
... Regarding the theory of symmetric spaces, there are a number of well written literature: Helgason [33,34], Knapp [46], and many more. The authors also have tried to describe the symmetric space with a linear algebra background in Section 3 of [23]. In this section we will briefly review the basics. ...
... Since this is one of the simplest KAK decompositions, it sometimes appeared in Lie theory literatures (e.g., [75]) as an example of the KAK decomposition. As a matrix factorization, the ODO decomposition has first appeared in [27], and the authors also discussed it in [23]. The ODO decomposition says that for any unitary matrix U , we have an identical set of (left and right) singular vectors for real and imaginary parts. ...
... The inverse matrix R −1 is just a block diagonal matrix of 2 × 2 rotation blocks cos θ l − sin θ l sin θ l cos θ l , where R has inverse rotation blocks cos θ l sin θ l − sin θ l cos θ l on its diagonal. The compact symmetric space U(2n)/ USp(2n) has the following KAK decomposition and we call this the QDQ decomposition [23]. • Two complexified n × n quaternionic unitary matrices ...
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.
The integral of a function f defined on a symmetric space \(M \simeq G/K\) may be expressed in the form of a determinant (or Pfaffian),when f is K-invariant and, in a certain sense, a tensor power of a positive function of a single variable. The paper presents a few examples of this idea and discusses future extensions. Specifically, the examples involve symmetric cones, Grassmann manifolds, and classical domains.
Keywordssymmetric spacematrix factorisationrandom matrices
