# positive definite

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##### 1: 35.1 Special Notation
 $a,b$ complex variables. … space of positive-definite real symmetric matrices. … $\mathbf{X}-\mathbf{T}$ is positive definite. Similarly, $\mathbf{T}<\mathbf{X}$ is equivalent. …
##### 2: 35.5 Bessel Functions of Matrix Argument
35.5.3 $B_{\nu}\left(\mathbf{T}\right)=\int_{\boldsymbol{\Omega}}\operatorname{etr}% \left(-(\mathbf{T}\mathbf{X}+\mathbf{X}^{-1})\right)\left|\mathbf{X}\right|^{% \nu-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},$ $\nu\in\mathbb{C}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.5.4 $\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \left|\mathbf{X}\right|^{\nu}A_{\nu}\left(\mathbf{S}\mathbf{X}\right)\,\mathrm% {d}{\mathbf{X}}=\operatorname{etr}\left(-\mathbf{S}\mathbf{T}^{-1}\right)\left% |\mathbf{T}\right|^{-\nu-\frac{1}{2}(m+1)},$ $\mathbf{S}\in\boldsymbol{\mathcal{S}}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$; $\Re\left(\nu\right)>-1$.
35.5.5 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}A_{\nu_{1}}\left(\mathbf{S% }_{1}\mathbf{X}\right)\left|\mathbf{X}\right|^{\nu_{1}}\*A_{\nu_{2}}\left(% \mathbf{S}_{2}(\mathbf{T}-\mathbf{X})\right)\left|\mathbf{T}-\mathbf{X}\right|% ^{\nu_{2}}\,\mathrm{d}{\mathbf{X}}=\left|\mathbf{T}\right|^{\nu_{1}+\nu_{2}+% \frac{1}{2}(m+1)}A_{\nu_{1}+\nu_{2}+\frac{1}{2}(m+1)}\left((\mathbf{S}_{1}+% \mathbf{S}_{2})\mathbf{T}\right),$ $\nu_{j}\in\mathbb{C}$, $\Re\left(\nu_{j}\right)>-1$, $j=1,2$; $\mathbf{S}_{1},\mathbf{S}_{2}\in\boldsymbol{\mathcal{S}}$; $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.5.6 $B_{\nu}\left(\mathbf{T}\right)=\left|\mathbf{T}\right|^{-\nu}B_{-\nu}\left(% \mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.5.7 $\int_{\boldsymbol{\Omega}}A_{\nu_{1}}\left(\mathbf{T}\mathbf{X}\right)B_{-\nu_% {2}}\left(\mathbf{S}\mathbf{X}\right)\left|\mathbf{X}\right|^{\nu_{1}}\,% \mathrm{d}{\mathbf{X}}=\frac{1}{A_{\nu_{1}+\nu_{2}}\left(\boldsymbol{{0}}% \right)}\left|\mathbf{S}\right|^{\nu_{2}}\left|\mathbf{T}+\mathbf{S}\right|^{-% (\nu_{1}+\nu_{2}+\frac{1}{2}(m+1))},$ $\Re\left(\nu_{1}+\nu_{2}\right)>-1$; $\mathbf{S},\mathbf{T}\in{\boldsymbol{\Omega}}$.
##### 3: 35.3 Multivariate Gamma and Beta Functions
35.3.1 $\Gamma_{m}\left(a\right)=\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-% \mathbf{X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\,\mathrm{d}{% \mathbf{X}},$ $\Re\left(a\right)>\frac{1}{2}(m-1)$.
35.3.2 $\Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}% \operatorname{etr}\left(-\mathbf{X}\right)\left|\mathbf{X}\right|^{s_{m}-\frac% {1}{2}(m+1)}\prod_{j=1}^{m-1}|(\mathbf{X})_{j}|^{s_{j}-s_{j+1}}\,\mathrm{d}{% \mathbf{X}},$ $s_{j}\in\mathbb{C}$, $\Re\left(s_{j}\right)>\frac{1}{2}(j-1)$, $j=1,\dots,m$.
35.3.8 $\mathrm{B}_{m}\left(a,b\right)=\int_{\boldsymbol{\Omega}}\left|\mathbf{X}% \right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}+\mathbf{X}\right|^{-(a+b)}\,% \mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.
##### 4: 19.31 Probability Distributions
More generally, let $\mathbf{A}$ ($=[a_{r,s}]$) and $\mathbf{B}$ ($=[b_{r,s}]$) be real positive-definite matrices with $n$ rows and $n$ columns, and let $\lambda_{1},\dots,\lambda_{n}$ be the eigenvalues of $\mathbf{A}\mathbf{B}^{-1}$. …
##### 5: 21.1 Special Notation
 $g,h$ positive integers. … $g\times g$ complex, symmetric matrix with $\Im\boldsymbol{{\Omega}}$ strictly positive definite, i.e., a Riemann matrix. …
##### 7: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.2 $\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m}\left(a\right)}\int_{% \boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}+\mathbf{X}\right|}^{% b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right)>\frac{1}{2}(m-1)$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.6.5 $\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \left|\mathbf{X}\right|^{b-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a\atop b};% \mathbf{S}\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}=\Gamma_{m}\left(b\right)% \left|\mathbf{I}-\mathbf{S}\mathbf{T}^{-1}\right|^{-a}\left|\mathbf{T}\right|^% {-b},$ $\mathbf{T}>\mathbf{S}$, $\Re\left(b\right)>\frac{1}{2}(m-1)$.
35.6.8 $\int_{\boldsymbol{\Omega}}\left|\mathbf{T}\right|^{c-\frac{1}{2}(m+1)}\Psi% \left(a;b;\mathbf{T}\right)\,\mathrm{d}{\mathbf{T}}=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(a-c\right)\Gamma_{m}\left(c-b+\frac{1}{2}(m+1)\right)}{% \Gamma_{m}\left(a\right)\Gamma_{m}\left(a-b+\frac{1}{2}(m+1)\right)},$ $\Re\left(a\right)>\Re\left(c\right)+\frac{1}{2}(m-1)>m-1$, $\Re\left(c-b\right)>-1$.
##### 8: 1.5 Calculus of Two or More Variables
and the second order term in (1.5.18) is positive definite (negative definite), that is, …
##### 9: 35.4 Partitions and Zonal Polynomials
35.4.8 $\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \,\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}Z_{\kappa}\left(\mathbf{X}\right% )\,\mathrm{d}{\mathbf{X}}=\Gamma_{m}\left(a+\kappa\right)\,\left|\mathbf{T}% \right|^{-a}Z_{\kappa}\left(\mathbf{T}^{-1}\right),$
##### 10: 18.34 Bessel Polynomials
The full system satisfies orthogonality with respect to a (not positive definite) moment functional; see Evans et al. (1993, (2.7)) for the simple expression of the moments $\mu_{n}$. …