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I have a positive definite Hermitian matrix $P=P^*>0$ where $P^*$ is the conjugate transpose of $P$ and $r=\exp(i\theta)$. So, how can I prove that $$ (rP+\bar{r}\bar{P})^{-1}=rY+\bar{r}\bar{Y}$$ Where, $Y$ is a positive definite Hermitian matrix and $\bar{Y}$ is its conjugate?

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