# Trace of integral of a function of two matrices

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.

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