triangular matrices

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31—40 of 43 matching pages

31: 32.4 Isomonodromy Problems

… ►is a linear system in which

\mathbf{A}

and

\mathbf{B}

are matrices and

\lambda

is independent of

z

. …

32: Bibliography G

… ►

P. Gianni, M. Seppälä, R. Silhol, and B. Trager (1998) Riemann surfaces, plane algebraic curves and their period matrices. J. Symbolic Comput. 26 (6), pp. 789–803.

ⓘ

External Links:: ISSN 0747-7171, Document, MathReview (Gabino González Díez), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

G. H. Golub and G. Meurant (2010) Matrices, moments and quadrature with applications. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ.

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External Links:: ISBN 978-0-691-14341-5, MathReview (G. A. Evans)
Referenced by:: §18.2(ix), §18.40(ii).
Encodings:: BibTeX

…

… ►For the computation of zeros of orthogonal polynomials as eigenvalues of finite tridiagonal matrices (§3.5(vi)), see Gil et al. (2007a, pp. 205–207). For the computation of zeros of Bessel functions, Coulomb functions, and conical functions as eigenvalues of finite parts of infinite tridiagonal matrices, see Grad and Zakrajšek (1973), Ikebe (1975), Ikebe et al. (1991), Ball (2000), and Gil et al. (2007a, pp. 205–213). …

34: Bibliography R

… ►

E. M. Rains (1998) Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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External Links:: ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

…

35: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

35.7.2

P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+% \nu+\frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*% {{}_{2}F_{1}}\left({-\nu,\gamma+\delta+\nu+\frac{1}{2}(m+1)\atop\gamma+\frac{1% }{2}(m+1)};\mathbf{T}\right),

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

;

\gamma,\delta,\nu\in\mathbb{C}

;

\Re\left(\gamma\right)>-1

ⓘ

Defines:: $P^{(\NVar{\gamma},\NVar{\delta})}_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Jacobi function of matrix argument
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, $\mathbb{C}$ : complex plane, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $<$ : less-than for matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(i), §35.7(i), §35.7 and Ch.35

… ►

35.7.5

{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left% (a,c-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}-\mathbf{X}\right|}^{% c-a-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{T}\mathbf{X}\right|}^{-b}\,% \mathrm{d}{\mathbf{X}},

\Re\left(a\right),\Re\left(c-a\right)>\frac{1}{2}(m-1)

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

ⓘ

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, $\int$ : integral, $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : 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, $b$ : complex variable, $<$ : less-than for matrices, $c$ : complex variable, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Referenced by:: §35.7(iv)
Permalink:: http://dlmf.nist.gov/35.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

…

36: Bibliography J

… ►

JTEM (website) Java Tools for Experimental Mathematics

ⓘ

Notes:: Single and double precision. Contains packages to compute elliptic functions, Riemann theta functions and their derivatives, and Riemann matrices for Riemann surfaces.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

…

37: Bibliography

… ►

J. Avron and B. Simon (1982) Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices. Bulletin of the American Mathematical Society 6 (1), pp. 81–85.

ⓘ

External Links:: MathReview Entry
Referenced by:: §1.18(vii).
Encodings:: BibTeX

…

38: 3.7 Ordinary Differential Equations

… ►(

\mathbf{I}

and

\boldsymbol{{0}}

being the identity and zero matrices of order

2\times 2

.) …

39: 18.38 Mathematical Applications

… ►Light and Carrington Jr. (2000) review and extend the one-dimensional analysis to solution of multi-dimensional many-particle systems, where the sparse nature of the resulting matrices is highly advantageous. …

40: 26.8 Set Partitions: Stirling Numbers

… ►Let

A

and

B

be the

n\times n

matrices with

(j,k)

th elements

s\left(j,k\right)

, and

S\left(j,k\right)

, respectively. …