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.
2026-03-25 06:21:16.1774419676
Trace of integral of a function of two matrices
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