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Can we find the sufficient and necessary condition of $a$, $b$ and $c$ $\in\mathbb R_+$ such that the following integration is integrable? $$ I_1\equiv\int \frac{1}{|\Sigma|^a|\Xi|^b|\mathrm{L}\Sigma\mathrm{L}'+\Xi|^c} \mathrm{etr}\left\{-\frac{1}{2} (\mathrm{L}\Sigma\mathrm{L}'+\Xi)^{-1}\mathrm{X}\right\} d\,\Sigma~ d\,\Xi\ , $$

where etr$(\mathrm{A})$ means exp$\{\mathrm tr(\mathrm{A})\}$. $\Sigma=diag\{\sigma_1,\cdots,\sigma_k\}$, $\Xi=diag\{\xi_1,\cdots,\xi_k\}$, $(\sigma_1,\cdots,\sigma_k)'\in\mathbb R^k_+$, $(\xi_1,\cdots,\xi_k)'\in\mathbb R^k_+$.

$\mathrm{X}$ is a nonnegative definite $k\times k$ matrix, $\mathrm{L}$ is a invertibe $k\times k$ matrix.

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  • $\begingroup$ I guess the boldface is to make it easier to identify matrices, but, when almost everything in sight is a matrix, it has the opposite effect (at least for me). $\endgroup$
    – LSpice
    Commented Jan 25, 2019 at 18:14
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    $\begingroup$ Yes, you are right, I have given up using \mathbf{ } to identify matrices. But the math notations still dsiplay as boldface, e.x., k\times k in mathmode displays as in bold ($k\times k$), which is not the form I want. I don't know what happened here. $\endgroup$
    – cong
    Commented Jan 25, 2019 at 18:41
  • $\begingroup$ It looks fine (i.e., not bolded) to me. $\endgroup$
    – LSpice
    Commented Jan 25, 2019 at 18:58

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