matrix elements of the resolvent

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11: 21.1 Special Notation

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$g,h$	positive integers.
…
$𝛀$	$g\times g$ complex, symmetric matrix with $\mathrm{\Im }𝛀$ strictly positive definite, i.e., a Riemann matrix.
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$A_{jk}$	$(j,k)$th element of matrix $𝐀$.
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${\mathrm{𝟎}}_g$	$g\times g$ zero matrix.
${𝐈}_g$	$g\times g$ identity matrix.
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$S_1{S}_2$	set of all elements of the form “$\text{<span class="ltx\_text hilite">element</span> of }{S}_1\times \text{<span class="ltx\_text hilite">element</span> of }{S}_2$”.
…

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12: 21.5 Modular Transformations

… ►is a symplectic matrix, that is, … ►($𝐀$ invertible with integer elements.) …($𝐁$ symmetric with integer elements and even diagonal elements.) …($𝐁$ symmetric with integer elements.) …For a $g\times g$ matrix $𝐀$ we define $\mathrm{diag}𝐀$, as a column vector with the diagonal entries as elements. …

13: 35.10 Methods of Computation

§35.10 Methods of Computation

… ►See Yan (1992) for the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $𝐎(m)$ applied to a generalization of the integral (35.5.8). …

14: 21.6 Products

… ►Let $𝐓=[T_{jk}]$ be an arbitrary $h\times h$ orthogonal matrix (that is, $𝐓{𝐓}^{\mathrm{T}}=𝐈$) with rational elements. Also, let $𝐙$ be an arbitrary $g\times h$ matrix. …that is, $𝒦$ is the set of all $g\times h$ matrices that are obtained by premultiplying $𝐓$ by any $g\times h$ matrix with integer elements; two such matrices in $𝒦$ are considered equivalent if their difference is a matrix with integer elements. …that is, $𝒟$ is the number of elements in the set containing all $h$-dimensional vectors obtained by multiplying ${𝐓}^{\mathrm{T}}$ on the right by a vector with integer elements. Two such vectors are considered equivalent if their difference is a vector with integer elements. …

15: 35.9 Applications

§35.9 Applications

►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 ${}_p{}^{}F_q^{}$, with $p\le 2$ and $q\le 1$. See James (1964), Muirhead (1982), Takemura (1984), Farrell (1985), and Chikuse (2003) for extensive treatments. … ►These references all use results related to the integral formulas (35.4.7) and (35.5.8). … ►In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. …

16: 35.4 Partitions and Zonal Polynomials

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35.4.3 $$Z_{\kappa }\left(𝐓\right)=Z_{\kappa }\left(𝐈\right){\left|𝐓\right|}^{k_m}\underset{𝐎(m)}{\int }\prod _{j=1}^{m-1}{|{(𝐇𝐓{𝐇}^{-1})}_j|}^{k_j-k_{j+1}}\mathrm{d}𝐇,$$ $𝐓\in 𝓢$.

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Symbols:

$\in$: element of, $\int$: integral, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

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35.4.5 $$Z_{\kappa }\left(𝐇𝐓{𝐇}^{-1}\right)=Z_{\kappa }\left(𝐓\right),$$ $𝐇\in 𝐎(m)$.

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Symbols:

$\in$: element of, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐓$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

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35.4.6 $$\sum _{|\kappa |=k}{Z}_{\kappa }\left(𝐓\right)={(\mathrm{tr}𝐓)}^k.$$

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Symbols:

$\mathrm{tr}𝐀$: trace of matrix, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐓$: real symmetric $m\times m$ matrix, $k$: nonnegative integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

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35.4.7 $${\int }_{𝐎(m)}{Z}_{\kappa }\left(𝐒𝐇𝐓{𝐇}^{-1}\right)\mathrm{d}𝐇=\frac{Z_{\kappa }\left(𝐒\right)Z_{\kappa }\left(𝐓\right)}{Z_{\kappa }\left(𝐈\right)}.$$

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Symbols:

$\int$: integral, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.9

Permalink:

http://dlmf.nist.gov/35.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►For $𝐓\in 𝛀$ and $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$, …

17: 35 Functions of Matrix Argument

Chapter 35 Functions of Matrix Argument

…

18: 26.15 Permutations: Matrix Notation

§26.15 Permutations: Matrix Notation

… ►The permutation $\sigma$ corresponds to the matrix in which there is a 1 at the intersection of row $j$ with column $\sigma (j)$, and 0’s in all other positions. The permutation $35247816$ corresponds to the matrix … ►The sign of the permutation $\sigma$ is the sign of the determinant of its matrix representation. The inversion number of $\sigma$ is a sum of products of pairs of entries in the matrix representation of $\sigma$: …

19: 35.3 Multivariate Gamma and Beta Functions

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35.3.1 $${\mathrm{\Gamma }}_m\left(a\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}d𝐗,$$ $\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$.

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Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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35.3.2 $${\mathrm{\Gamma }}_m\left(s_1,\mathrm{\dots },s_m\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{s_m-\frac{1}{2}(m+1)}\prod _{j=1}^{m-1}{|{(𝐗)}_j|}^{s_j-s_{j+1}}d𝐗,$$ $s_j\in ℂ$, $\mathrm{\Re }\left(s_j\right)>\frac{1}{2}(j-1)$, $j=1,\mathrm{\dots },m$.

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $j$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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35.3.3 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

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Defines:

${\mathrm{B}}_m(a,b)$: multivariate beta function

Symbols:

$\int$: integral, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Keywords:

definition, multivariate beta function

Permalink:

http://dlmf.nist.gov/35.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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35.3.8 $${\mathrm{B}}_m(a,b)={\int }_{𝛀}{\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{\left|𝐈+𝐗\right|}^{-(a+b)}d𝐗,$$ $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

$\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

20: 21.3 Symmetry and Quasi-Periodicity

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21.3.1 $$\theta \left(-𝐳\right|𝛀)=\theta \left(𝐳\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.3(i), §21.3 and Ch.21

… ►when ${𝐦}_1\in {\mathbb{Z}}^g.$ Thus $\theta \left(𝐳\right|𝛀)$ is periodic, with period $1$, in each element of $𝐳$. …with ${𝐦}_1$, ${𝐦}_2$ $\in {\mathbb{Z}}^g$. … ►Again, with ${𝐦}_1$, ${𝐦}_2$ $\in {\mathbb{Z}}^g$ …Because of this property, the elements of $𝜶$ and $𝜷$ are usually restricted to $[0,1)$, without loss of generality. …