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> Is there any integral expression for $\log (X + Y) - \log (X)$ if $X$ and $Y$ are positive definite matrices?
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Could anyone give some suggestion as to how to find such an integral expression if there is any?


Thanks a bunch.



Question on integral expression of positive definite matrices
The [formula](https://math.stackexchange.com/a/2085547/87355)
$$\frac{d}{ds}\log Z(s) = \int\_0^1 [(1-t)I+tZ(s)]^{-1}Z'(s) [(1-t)I+tZ(s)]^{-1}\, dt,$$
with $Z(s)=X+sY$, gives upon integration of
$$\int\_0^1 \frac{d}{ds}\log Z(s)\,ds=\log Z(1)-\log Z(0)$$
an integral expression for
$$\log(X+Y)-\log X=\int\_0^1\int\_0^1 [(1-t)I+t(X+sY)]^{-1} Y [(1-t)I+t(X+sY)]^{-1}\, dt\,ds.$$


