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31—40 of 74 matching pages

31: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

1.3.6

\det\left({\mathbf{A}}^{-1}\right)=\frac{1}{\det(\mathbf{A})},

ⓘ

Symbols:: $\det$ : determinant, ${\NVar{\mathbf{A}}}^{-1}$ : matrix inverse, $j$ : integer and $k$ : integer
Referenced by:: Erratum (V1.2.0) for Equations (1.3.5), (1.3.6), (1.3.7)
Permalink:: http://dlmf.nist.gov/1.3.E6
Encodings:: pMML, png, TeX
Modification (effective with 1.2.0):: Previously we did use the notation $[a_{jk}]$ for $\mathbf{A}$ .
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

… ►The determinant of an upper or lower triangular, or diagonal, square matrix

\mathbf{A}

is the product of the diagonal elements

\det(\mathbf{A})=\prod_{i=1}^{n}a_{ii}

. … ►The adjoint of a matrix

\mathbf{A}

is the matrix

{\mathbf{A}}^{*}

such that

\left\langle\mathbf{A}\mathbf{a},\mathbf{b}\right\rangle=\left\langle\mathbf{a% },{\mathbf{A}}^{*}\mathbf{b}\right\rangle

for all

\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}

. In the case of a real matrix

{\mathbf{A}}^{*}=\mathbf{A}^{\mathrm{T}}

and in the complex case

{\mathbf{A}}^{*}={\mathbf{A}}^{{\rm H}}

. … ►Let the columns of matrix

\mathbf{S}

be these eigenvectors

\mathbf{a}_{1},\dots,\mathbf{a}_{n}

, then

{\mathbf{S}}^{-1}={\mathbf{S}}^{{\rm H}}

, and the similarity transformation (1.2.73) is now of the form

{\mathbf{S}}^{{\rm H}}\mathbf{A}\mathbf{S}=\lambda_{i}\delta_{i,j}

. …

32: 19.31 Probability Distributions

… ►

19.31.1

\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x}=\sum_{r=1}^{n}\sum_{s=1}^{n}a_{r,s% }x_{r}x_{s},

ⓘ

Symbols:: $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/19.31.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.31 and Ch.19

… ►

19.31.2

\int_{{\mathbb{R}}^{n}}(\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x})^{\mu}\exp% \left(-\mathbf{x}^{\mathrm{T}}\mathbf{B}\mathbf{x}\right)\,\mathrm{d}x_{1}% \cdots\,\mathrm{d}x_{n}=\frac{\pi^{n/2}\Gamma\left(\mu+\tfrac{1}{2}n\right)}{% \sqrt{\det\mathbf{B}}\Gamma\left(\tfrac{1}{2}n\right)}R_{\mu}\left(\tfrac{1}{2% },\dots,\tfrac{1}{2};\lambda_{1},\dots,\lambda_{n}\right),

\mu>-\tfrac{1}{2}n

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\mathbb{R}$ : real line, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $n$ : nonnegative integer and $\lambda_{n}$ : eigenvalues
Referenced by:: §19.31
Permalink:: http://dlmf.nist.gov/19.31.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.31 and Ch.19

…

33: Bibliography Y

… ►

H. A. Yamani and L. Fishman (1975)

J

-matrix method: Extensions to arbitrary angular momentum and to Coulomb scattering. J. Math. Phys. 16, pp. 410–420.

ⓘ

Referenced by:: §18.39(iv).
Encodings:: BibTeX

…

… ►At CalTech, John Todd dedicated himself to the training of new researchers in numerical analysis, and Olga Taussky, who had been a full-time NBS consultant influential in establishing the field of matrix theory, became the first woman at CalTech to attain the academic rank of full professor. … ►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. …

35: Mathematical Introduction

… ► ►►►►►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$[a_{j,k}]$ or $[a_{jk}]$	matrix with $(j,k)$ th element $a_{j,k}$ or $a_{jk}$ .
${\mathbf{A}}^{-1}$	inverse of matrix $\mathbf{A}$ .
$\operatorname{tr}\mathbf{A}$	trace of matrix $\mathbf{A}$ .
$\mathbf{A}^{\mathrm{T}}$	transpose of matrix $\mathbf{A}$ .
$\mathbf{I}$	unit matrix.
…

…

36: 18.36 Miscellaneous Polynomials

… ►

§18.36(iv) Orthogonal Matrix Polynomials

►These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second order matrix differential equations with coefficients independent of the degree. …

37: 15.17 Mathematical Applications

… ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL

(2,\mathbb{R})

, and spherical functions on certain nonsymmetric Gelfand pairs. …

38: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►Let

\alpha=-n

n=0,1,2,\dots

, and

q_{n,m}

m=0,1,\dots,n

, be the eigenvalues of the tridiagonal matrix …

39: 18.38 Mathematical Applications

… ►

Random Matrix Theory

►Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. …

40: 21.9 Integrable Equations

… ►

21.9.4

u(x,y,t)=c+2\frac{{\partial}^{2}}{{\partial x}^{2}}\ln\left(\theta\left(% \mathbf{k}x+\mathbf{l}y+\boldsymbol{{\omega}}t+\boldsymbol{{\phi}}\middle|% \boldsymbol{{\Omega}}\right)\right),

ⓘ

Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\ln\NVar{z}$ : principal branch of logarithm function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\boldsymbol{{\Omega}}$ : a Riemann matrix, $u(x,y,t)$ : elevation and $c$ : complex constant
Referenced by:: §21.9
Permalink:: http://dlmf.nist.gov/21.9.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §21.9 and Ch.21

… ►Furthermore, the solutions of the KP equation solve the Schottky problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemann surface (Schottky (1903)). …