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Suppose $B$ is a positive definite matrix with determinant $1 $ and $$ A = \frac{1}{2} \int_0^\infty \frac{(B+sI)^{-1}}{\sqrt{\mbox{det}(B+sI)}} ds $$

Then, how does one prove that this provides a one to one onto correspondence between positive definite matrices $B$ with determinant $1$ and positive definite matrices $A$ with trace $1$.

Thank you very much.

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    Trace and determinant are invariants under coordinate changes and positive definite matrices are diagonalizable, so try diagonalizing $B$. After shifting to diagonals, I think the sum of the integrals on the diagonal of $A$ should collapse down some and leave you with a manageable computation. Maybe someone else has a better (i.e. slick) way to proceed.2011-11-08
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    Thanks for the hint. How does one show that this mapping is one to one and onto though.2011-11-08
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    My comment just suggests how to compute with this formula. It certainly doesn't address whether the transformation is bijective. Where is this map from? I'm not familiar with it. Maybe some context will help.2011-11-09

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