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TRIDIAGONAL REALIZATION OF THE ANTISYMMETRIC GAUSSIAN βENSEMBLE
, 904
"... Abstract. The Householder reduction of a member of the antisymmetric Gaussian unitary ensemble gives an antisymmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter β, and the eigenvalue probability density function of the cor ..."
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Abstract. The Householder reduction of a member of the antisymmetric Gaussian unitary ensemble gives an antisymmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter β, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of {qi}, the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the antisymmetric tridiagonal matrices. This proof uses the DixonAnderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real antisymmetric tridiagonal matrices, its eigenvalues and {qi}. The third proof, which is restricted to n even, maps matrices from the antisymmetric Gaussian βensemble to those realizing particular examples of the Laguerre βensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Prüfer phases of the random matrices. 1.
Random matrix theory and its innovative applications
 Advances in Applied Mathematics, Modeling, and Computational Science 66
, 2013
"... 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 exper ..."
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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. 1 Random Matrix Theory in the Press Since the beginning of the 20th century, Random matrix theory (RMT) has been finding applications in number theory, quantum mechanics, condensed matter physics, wireless communications, etc., see [16, 15, 12, 7]. Recently more and more disciplines of science and engineering have found RMT valuable. New applications in RMT are being found every day, some of them surprising and innovative when compared with the older applications. For newcomers to the field, it may be reassuring to know that very little specialized knowledge of random matrix theory is required for applications, and therefore the “learning curve ” to become a user is not at all steep. Two methodologies are worth highlighting.