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I have two positive definite matrices $X$ and $Y$, such that $\|X\|<1$ and $\|Y\|<1$. I have a trace of an integral of the form: $$\text{trace}\left(\int_0^{\infty}X(Y+t)^{-1}X(Y+t)^{-1}dt\right)$$ I am interested to turn it to trace of some function of $X$ and $Y$ without the integral, for example something like $\text{trace} (X^{2}Y^{-1})$. Is this possible to do that? I have no idea if it is even possible? any help is very appreciated.

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    $\begingroup$ have you tried with a 2x2 example? $\endgroup$ – thedude Feb 5 '18 at 15:51
  • $\begingroup$ Tried it with $2\times 2$ matrices. Turned out that they are not equal. Thanks! $\endgroup$ – Mah Feb 5 '18 at 17:57
  • $\begingroup$ is it $t$ multiplying by the identity? $\endgroup$ – Carlos Campos Feb 5 '18 at 18:23
  • $\begingroup$ @CarlosCampos Yes. It is multiplied by the identity matrix. $\endgroup$ – Mah Feb 5 '18 at 18:29
  • $\begingroup$ It seems there no exists a closed form for $(A+I)^{-1}$: math.stackexchange.com/questions/298616/what-is-inverse-of-ia $\endgroup$ – Carlos Campos Feb 5 '18 at 18:39

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