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In a paper by Mathai, he uses the following integrap integral representation of a determinant, (or, really, what I give is a simple special case of what he gives), without any explication. All matrices are real $p\times p$ symmetric positive definite.

$$| I-U |^{-a} = \frac{1}{\Gamma_p(a)} \int_{T>0} |T|^{a-(p+1)/2} \exp(-\text{Tr}(I-U)T) \;dT$$

where $U$ satisfies $0(p-1)/2$.0\lt U \lt I$(in the cone order in the cone of positive definte matrices), the integral is over the cone of positive definite matrices and$\Gamma_p(a)$is the generalized gamma function in dimension$p$, and$\Re(a) > (p-1)/2$. Any references for this? 1 # Integral representation of a determinant In a paper by Mathai, he uses the following integrap representation of a determinant, (or, really, what I give is a simple special case of what he gives), without any explication. All matrices are real$p\times p$symmetric positive definite. $$| I-U |^{-a} = \frac{1}{\Gamma_p(a)} \int_{T>0} |T|^{a-(p+1)/2} \exp(-\text{Tr}(I-U)T) \;dT$$ where$U$satisfies$0(p-1)/2\$.

Any references for this?