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21: 28.34 Methods of Computation

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(d)

Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

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22: Philip J. Davis

… ►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. …

23: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.2

{{}_{0}F_{0}}\left({-\atop-};\mathbf{T}\right)=\operatorname{etr}\left(\mathbf% {T}\right),

\mathbf{T}\in\boldsymbol{\mathcal{S}}

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\in$ : element of, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\boldsymbol{\mathcal{S}}$ : space of real symmetric $m\times m$ matrices and $\mathbf{T}$ : real symmetric $m\times m$ matrix
Permalink:: http://dlmf.nist.gov/35.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(ii), §35.8 and Ch.35

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35.8.3

{{}_{2}F_{1}}\left({a,b\atop b};\mathbf{T}\right)={{}_{1}F_{0}}\left({a\atop-}% ;\mathbf{T}\right)=\left|\mathbf{I}-\mathbf{T}\right|^{-a},

\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $b$ : complex variable, $<$ : less-than for matrices and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(ii), §35.8 and Ch.35

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35.8.4

A_{\nu}\left(\mathbf{T}\right)=\dfrac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(m+1)% \right)}{{}_{0}F_{1}}\left({-\atop\nu+\frac{1}{2}(m+1)};-\mathbf{T}\right),

\mathbf{T}\in\boldsymbol{\mathcal{S}}

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Symbols:: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\boldsymbol{\mathcal{S}}$ : space of real symmetric $m\times m$ matrices, $\mathbf{T}$ : real symmetric $m\times m$ matrix and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(ii), §35.8 and Ch.35

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35.8.11

{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{H}% \mathbf{T}\mathbf{H}^{-1}\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_% {1},\dots,b_{q}};\mathbf{T}\right),

\mathbf{H}\in\mathbf{O}(m)

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\in$ : element of, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $\mathbf{O}(m)$ : space of $m\times m$ orthogonal matrices, $\mathbf{H}$ : orthogonal $m\times m$ matrix, $p$ : nonnegative integer, $q$ : nonnegative integer and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.8.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

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35.8.12

{\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right% )\left|\mathbf{X}\right|^{\gamma-\frac{1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};-\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}% }=\Gamma_{m}\left(\gamma\right)\left|\mathbf{T}\right|^{-\gamma}{{}_{p+1}F_{q}% }\left({a_{1},\dots,a_{p},\gamma\atop b_{1},\dots,b_{q}};-\mathbf{T}^{-1}% \right),

\Re\left(\gamma\right)>\frac{1}{2}(m-1)

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : complex variable, $p$ : nonnegative integer, $q$ : nonnegative integer and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.8.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

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24: Bibliography F

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P. J. Forrester and N. S. Witte (2001) Application of the

\tau

-function theory of Painlevé equations to random matrices: PIV, PII and the GUE. Comm. Math. Phys. 219 (2), pp. 357–398.

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External Links:: ISSN 0010-3616, Document, MathReview (Alexei Khorunzhy), ZentralBlatt
Referenced by:: §32.10(iv), §32.14, §32.6(v).
Encodings:: BibTeX

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P. J. Forrester and N. S. Witte (2002) Application of the

\tau

-function theory of Painlevé equations to random matrices:

\mathrm{P}_{\mathrm{V}}

\mathrm{P}_{\mathrm{III}}

, the LUE, JUE, and CUE. Comm. Pure Appl. Math. 55 (6), pp. 679–727.

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External Links:: ISSN 0010-3640, Document, MathReview (Mark Adler), ZentralBlatt
Referenced by:: §32.10(iii), §32.10(v), §32.14, §32.6(iv), §32.6(v).
Encodings:: BibTeX

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P. J. Forrester and N. S. Witte (2004) Application of the

\tau

-function theory of Painlevé equations to random matrices:

\rm P_{\rm VI}

, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

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External Links:: ISSN 0027-7630, MathReview, ZentralBlatt
Referenced by:: §32.10(vi), §32.6(v).
Encodings:: BibTeX

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25: Bibliography I

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Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of

J_{0}(z)-iJ_{1}(z)

and of Bessel functions

J_{m}(z)

of any real order

m

. Linear Algebra Appl. 194, pp. 35–70.

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External Links:: ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt
Referenced by:: §10.21(ix).
Encodings:: BibTeX

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26: 34.3 Basic Properties: $\mathit{3j}$ Symbol

… ►Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to

\mathit{3j}

symbols, for which see Edmonds (1974, Chapter 4). …

27: Bibliography D

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B. Deconinck and M. van Hoeij (2001) Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.

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External Links:: ISSN 0167-2789, Document, MathReview (John B. Little), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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P. A. Deift (1998) Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, Vol. 3, New York University Courant Institute of Mathematical Sciences, New York.

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External Links:: ISBN 0-9658703-2-4; 0-8218-2695-6, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §18.38(ii), §18.38(ii).
Encodings:: BibTeX

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P. Di Francesco, P. Ginsparg, and J. Zinn-Justin (1995)

2

D gravity and random matrices. Phys. Rep. 254 (1-2), pp. 1–133.

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External Links:: ISSN 0370-1573, Document, MathReview (Yolanda Lozano)
Referenced by:: §32.16.
Encodings:: BibTeX

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28: 3.4 Differentiation

… ►For additional formulas involving values of

\nabla^{2}u

and

\nabla^{4}u

on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546). …

29: 21.5 Modular Transformations

… ►Let

\mathbf{A}

\mathbf{B}

\mathbf{C}

, and

\mathbf{D}

g\times g

matrices with integer elements such that …

30: 26.15 Permutations: Matrix Notation

… ►The set

\mathfrak{S}_{n}

(§26.13) can be identified with the set of

n\times n

matrices of 0’s and 1’s with exactly one 1 in each row and column. …