exponential of the trace

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1: 35.2 Laplace Transform

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35.2.1 $$g(𝐙)={\int }_{𝛀}\mathrm{etr}\left(-𝐙𝐗\right)f(𝐗)d𝐗,$$

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Defines:

$g$: Laplace transformation of $f$ (locally)

Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $𝐙$: complex symmetric $m\times m$ matrix and $f(𝐗)$: complex-valued function of matrix argument

Keywords:

Laplace transform

Referenced by:

§35.2

Permalink:

http://dlmf.nist.gov/35.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.2, §35.2 and Ch.35

… ►Suppose there exists a constant ${𝐗}_0\in 𝛀$ such that for all $𝐗\in 𝛀$. … ►

35.2.2 $$f(𝐗)=\frac{1}{{(2\pi \mathrm{i})}^{m(m+1)/2}}\int \mathrm{etr}\left(𝐙𝐗\right)g(𝐙)d𝐙,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\mathrm{i}$: imaginary unit, $\int$: integral, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, $𝐙$: complex symmetric $m\times m$ matrix, $f(𝐗)$: complex-valued function of matrix argument, $m$: positive integer and $g$: Laplace transformation of $f$

Permalink:

http://dlmf.nist.gov/35.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.2, §35.2 and Ch.35

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2: 35.5 Bessel Functions of Matrix Argument

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35.5.3 $$B_{\nu }\left(𝐓\right)={\int }_{𝛀}\mathrm{etr}\left(-(𝐓𝐗+{𝐗}^{-1})\right){\left|𝐗\right|}^{\nu -\frac{1}{2}(m+1)}d𝐗,$$ $\nu \in ℂ$, $𝐓\in 𝛀$.

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Defines:

$B_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (second kind)

Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐓$: real symmetric $m\times m$ 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 and $m$: positive integer

Referenced by:

§35.9

Permalink:

http://dlmf.nist.gov/35.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(i), §35.5 and Ch.35

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35.5.4 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{\nu }{A}_{\nu }\left(𝐒𝐗\right)d𝐗=\mathrm{etr}\left(-𝐒{𝐓}^{-1}\right){\left|𝐓\right|}^{-\nu -\frac{1}{2}(m+1)},$$ $𝐒\in 𝓢$, $𝐓\in 𝛀$; $\mathrm{\Re }\left(\nu \right)>-1$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $\mathrm{\Re }$: real part, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ 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 and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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35.5.8 $${\int }_{𝐎(m)}\mathrm{etr}\left(𝐒𝐇\right)\mathrm{d}𝐇=\frac{A_{-1/2}\left(-\frac{1}{4}𝐒{𝐒}^{\mathrm{T}}\right)}{A_{-1/2}\left(𝟎\right)},$$ $𝐒$ arbitrary.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐒$: 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 $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.10, §35.9

Permalink:

http://dlmf.nist.gov/35.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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3: 1.1 Special Notation

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$x,y$	real variables.
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$\mathrm{etr}\left(𝐀\right)$	exponential of $\mathrm{tr}(𝐀)$
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4: 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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5: 35.6 Confluent Hypergeometric Functions of Matrix Argument

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35.6.2 $$\mathrm{\Psi }(a;b;𝐓)=\frac{1}{{\mathrm{\Gamma }}_m\left(a\right)}{\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{\left|𝐈+𝐗\right|}^{b-a-\frac{1}{2}(m+1)}d𝐗,$$ $\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$, $𝐓\in 𝛀$.

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Defines:

$\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind)

Symbols:

$\in$: 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, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ 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/35.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(i), §35.6 and Ch.35

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35.6.4 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b-a\right)>\frac{1}{2}(m-1)$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ 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

Permalink:

http://dlmf.nist.gov/35.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

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35.6.5 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{b-\frac{1}{2}(m+1)}{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{b};𝐒𝐗)d𝐗={\mathrm{\Gamma }}_m\left(b\right){\left|𝐈-𝐒{𝐓}^{-1}\right|}^{-a}{\left|𝐓\right|}^{-b},$$ $𝐓>𝐒$, $\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ 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, $>$: greater-than for matrices and $m$: positive integer

Referenced by:

§35.6(ii)

Permalink:

http://dlmf.nist.gov/35.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

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35.6.7 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{b};𝐓)=\mathrm{etr}\left(𝐓\right){}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{b-a}{b};-𝐓).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix and $b$: complex variable

Permalink:

http://dlmf.nist.gov/35.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

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6: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.2 $${}_0{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-}{-};𝐓)=\mathrm{etr}\left(𝐓\right),$$ $𝐓\in 𝓢$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $𝓢$: space of real symmetric $m\times m$ matrices and $𝐓$: 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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35.8.12 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{\gamma -\frac{1}{2}(m+1)}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};-𝐗)d𝐗={\mathrm{\Gamma }}_m\left(\gamma \right){\left|𝐓\right|}^{-\gamma }{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,\gamma }{b_1,\mathrm{\dots },b_q};-{𝐓}^{-1}),$$ $\mathrm{\Re }\left(\gamma \right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: real symmetric $m\times m$ 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, $p$: 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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7: 35.4 Partitions and Zonal Polynomials

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35.4.8 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{Z}_{\kappa }\left(𝐗\right)d𝐗={\mathrm{\Gamma }}_m\left(a+\kappa \right){\left|𝐓\right|}^{-a}{Z}_{\kappa }\left({𝐓}^{-1}\right),$$

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Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ 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, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E8

Encodings:

pMML, png, TeX

See also:

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

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8: 35.1 Special Notation

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$a,b$	complex variables.
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$\mathrm{etr}\left(𝐗\right)$	$\exp \left(\mathrm{tr}𝐗\right)$.
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9: 1.2 Elementary Algebra

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The Matrix Exponential and the Exponential of the Trace

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1.2.77 $$det(\exp \left(𝐀\right))=\exp \left(\mathrm{tr}(𝐀)\right)=\mathrm{etr}\left(𝐀\right).$$

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Defines:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace

Symbols:

$det$: determinant, $\exp z$: exponential function, $\mathrm{tr}𝐀$: trace of matrix and $𝐀$: non-defective matrix

Referenced by:

§1.2(vi), Erratum (V1.2.0) Section 1.2

Permalink:

http://dlmf.nist.gov/1.2.E77

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

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10: 21.6 Products

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21.6.3 $$\prod _{j=1}^h\theta \left(\sum _{k=1}^h{T}_{jk}{𝐳}_k\right|𝛀)=\frac{1}{{𝒟}^g}\sum _{𝐀\in 𝒦}\sum _{𝐁\in 𝒦}{\mathrm{e}}^{2\pi \mathrm{i}\mathrm{tr}\left[\frac{1}{2}{𝐀}^{\mathrm{T}}𝛀𝐀+{𝐀}^{\mathrm{T}}[𝐙+𝐁]\right]}\prod _{j=1}^h\theta \left({𝐳}_j+𝛀{𝐚}_j+{𝐛}_j\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{tr}𝐀$: trace of matrix, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $g$: positive integer, $h$: positive integer, $𝛀$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements

Permalink:

http://dlmf.nist.gov/21.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

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