Let $X$, $Y$ and $Z$ be positive definite matrices. How can we write the following inequality as an LMI?
$$XY - Z^2 - I \succ 0$$
Here, $I$ is the identity matrix.
For example, if it was $XY-Z^2Y-I>0$, its corresponding LMI would be readily accessible by:
$$ (X-Z^2)Y-I>0 \ \ \Big(\text{assuming that $X-Z^2>0$}\Big) \to\begin{bmatrix}X-Z^2 &I\\I &Y\end{bmatrix}>0 $$ $$ \to \begin{bmatrix}X & I & Z\\I & Y & 0\\Z &0&I\end{bmatrix}>0 $$ which is linear in terms of $X$, $Y$ and $Z$.