rotation matrices
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1: Philip J. Davis
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►After receiving an overview of the project and watching a short demo that included a few preliminary colorful, but static, 3D graphs constructed for the first Chapter, “Airy and Related Functions”, written by Olver, Davis expressed the hope that designing a web-based resource would allow the team to incorporate interesting computer graphics, such as function surfaces that could be rotated and examined.
This immediately led to discussions among some of the project members about what might be possible, and the discovery that some interactive graphics work had already been done for the NIST Matrix Market, a publicly available repository of test matrices for comparing the effectiveness of numerical linear algebra algorithms.
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►DLMF users can rotate, rescale, zoom and otherwise explore mathematical function surfaces.
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2: 13.27 Mathematical Applications
§13.27 Mathematical Applications
►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. …Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. … …3: 34.3 Basic Properties: Symbol
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►Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to symbols, for which see Edmonds (1974, Chapter 4).
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4: 35.1 Special Notation
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►All matrices are of order , unless specified otherwise.
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complex variables. | |
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space of all real symmetric matrices. | |
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space of positive-definite real symmetric matrices. | |
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is positive definite. Similarly, is equivalent. | |
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space of orthogonal matrices. | |
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5: 29.20 Methods of Computation
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►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv).
These matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that has to be chosen sufficiently large.
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►The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices
given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998).
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6: 35.9 Applications
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►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument , with and .
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►In the nascent area of applications of zonal polynomials to the limiting probability distributions of symmetric random matrices, one of the most comprehensive accounts is Rains (1998).
7: Viewing DLMF Interactive 3D Graphics
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►Users can render a 3D scene and interactively rotate, scale, and otherwise explore a function surface.
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8: Morris Newman
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►Newman wrote the book Matrix Representations of Groups, published by the National Bureau of Standards in 1968, and the book Integral Matrices, published by Academic Press in 1972, which became a classic.
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9: 32.14 Combinatorics
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►The distribution function given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of Hermitian matrices; see Tracy and Widom (1994).
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