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Scaling for orthogonality
Full text linkEdelman, Alan; Stewart, G.W.. (1992). Scaling for orthogonality. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/1905
Random Matrix Theory and Its Innovative Applications
Full text linkRecently more and more disciplines of science and engineering have found Random Matrix Theory valuable. Some disciplines use the limiting densities to indicate the cutoff between "noise" and "signal." Other disciplines are finding eigenvalue repulsions a compelling model of reality. This survey introduces both the theory behind these applications and MATLAB experiments allowing a reader immediate access to the ideas. Keywords:
computational science; dynamic blocking problems; elliptic curves; mathematical modeling; random matrix theor
Condition numbers of indefinite rank 2 ghost Wishart matrices
Full text linkAbstract We define an indefinite Wishart matrix as a matrix of the form A= W[superscript T]WΣ, where Σ is an indefinite diagonal matrix and W is a matrix of independent standard normals. We focus on the case where W is L×2 which has engineering applications. We obtain the distribution of the ratio of the eigenvalues of A. This distribution can be "folded" to give the distribution of the condition number. We calculate formulas for W real (β=1), complex (β=2), quaternionic (β=4) or any ghost 0 < β < ∞. We then corroborate our work by comparing them against numerical experiments.National Science Foundation (U.S.) (Grant CCF-0829421)National Science Foundation (U.S.) (SOLAR Grant 1035400)National Science Foundation (U.S.) (Grant DMS-1035400
Random Triangle Theory with Geometry and Applications
No full textWhat is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis, random matrix theory, the Hopf fibration, and much more. One of the best distributions of random triangles takes all six vertex coordinates as independent standard Gaussians. Six can be reduced to four by translation of the center to (0,0) or reformulation as a 2 × 2 random matrix problem. In this note, we develop shape theory in its historical context for a wide audience. We hope to encourage others to look again (and differently) at triangles. We provide a new constructive proof, using the geometry of parallelians, of a central result of shape theory: triangle shapes naturally fall on a hemisphere. We give several proofs of the key random result: that triangles are uniformly distributed when the normal distribution is transferred to the hemisphere. A new proof connects to the distribution of random condition numbers. Generalizing to higher dimensions, we obtain the “square root ellipticity statistic” of random matrix theory. Another proof connects the Hopf map to the SVD of 2 × 2 matrices. A new theorem describes three similar triangles hidden in the hemisphere. Many triangle properties are reformulated as matrix theorems, providing insight into both. This paper argues for a shift of viewpoint to the modern approaches of random matrix theory. As one example, we propose that the smallest singular value is an effective test for uniformity. New software is developed, and applications are proposed.National Science Foundation (U.S.) (NSF DMS 1035400)National Science Foundation (U.S.) (NSF DMS 1016125)National Science Foundation (U.S.) (NSF EFRI 1023152
Infinite random matrix theory, tridiagonal bordered Toeplitz matrices, and the moment problem
No full textThe four major asymptotic level density laws of random matrix theory may all be showcased through their Jacobi parameter representation as having a bordered Toeplitz form. We compare and contrast these laws, completing and exploring their representations in one place. Inspired by the bordered Toeplitz form, we propose an algorithm for the finite moment problem by proposing a solution whose density has a bordered Toeplitz form.National Science Foundation (U.S.) (DMS-1312831, DMS-1016125, DMS-1016086
The Beta-MANOVA Ensemble with General Covariance
Full text linkWe find the joint generalized singular value distribution and largest generalized singular value distributions of the β -MANOVA ensemble with positive diagonal covariance, which is general. This has been done for the continuous β > 0 case for identity covariance (in eigenvalue form), and by setting the covariance to I in our model we get another version. For the diagonal covariance case, it has only been done for β = 1, 2, 4 cases (real, complex, and quaternion matrix entries). This is in a way the first second-order β-ensemble, since the sampler for the generalized singular values of the β-MANOVA with diagonal covariance calls the sampler for the eigenvalues of the β-Wishart with diagonal covariance of Forrester and Dubbs-Edelman-Koev-Venkataramana. We use a conjecture of MacDonald proven by Baker and Forrester concerning an integral of a hypergeometric function and a theorem of Kaneko concerning an integral of Jack polynomials to derive our generalized singular value distributions. In addition we use many identities from Forrester’s Log-Gases and Random Matrices. We supply numerical evidence that our theorems are correct
Eigenvalue distributions of beta-Wishart matrices
Full text linkWe derive explicit expressions for the distributions of the extreme eigenvalues of the Beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results generalize the classical results for the real (β = 1), complex (β = 2), and quaternion (β = 4) Wishart matrices to any β > 0
Random matrix theory, numerical computation and applications
No full textThis paper serves to prove the thesis that a computational trick can open entirely new approaches to theory. We illustrate this by describ- ing such random matrix techniques as the stochastic operator approach, the method of ghosts and shadows, and the method of “Riccatti Diffusion/Sturm Sequences.” We thereby provide new insights into the deeper mathematics underlying random matrix theory
The singular values of the GUE (less is more)
Full text linkSome properties that nominally involve the eigenvalues of Gaussian Unitary
Ensemble (GUE) can instead be phrased in terms of singular values. By
discarding the signs of the eigenvalues, we gain access to a surprising
decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This
independence is remarkable given the well known phenomenon of eigenvalue repulsion. The structure of this decomposition reveals that several existing observations about large n limits of the GUE are in fact manifestations of phenomena that are already present for finite random matrices. We relate the semicircle law to the quarter-circle law by connecting Hermite polynomials to generalized Laguerre polynomials with parameter ± 1/2. Similarly, we write the absolute value of the determinant of the n x n GUE as a product n independent random variables to gain new insight into its asymptotic log-normality. The decomposition also provides a description of the distribution of the smallest singular value of the GUE, which in turn permits the study of the leading order behavior of the condition number of GUE matrices. The study is motivated by questions involving the enumeration of orientable maps, and is related to questions involving powers of complex Ginibre matrices. The inescapable conclusion of this work is that the singular values of the GUE play an unpredictably important role that had gone unnoticed for decades even though, in hindsight, so many clues had been around.National Science Foundation (U.S.) (Grant DMS–1035400)National Science Foundation (U.S.) (Grant DMS–1016125
Beyond universality in random matrix theory
No full textIn order to have a better understanding of finite random matrices with non-Gaussian entries, we study the 1/N expansion of local eigenvalue statistics in both the bulk and at the hard edge of the spectrum of random matrices. This gives valuable information about the smallest singular value not seen in universality laws. In particular, we show the dependence on the fourth moment (or the kurtosis) of the entries. This work makes use of the so-called complex Gaussian divisible ensembles for both Wigner and sample covariance matrices
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