Given a matrix $A\in\mathbb{C}^{n\times n}$, such that $\min\{\mathcal{R}(\lambda_i):\lambda_i~\text{is an eigenvalue of $A$}~\}>0$ is non-defective. Let $z_i$ be the (column) eigenvectors of $A$, and define
$$Z:=\begin{pmatrix}z_1&\ldots&z_n\end{pmatrix}.$$
What properties/can we say anything obvious about the product $$A^{-1}ZZ^T$$ have? Note that $AZ=\Lambda Z$ where $\Lambda$ is the diagonal matrix of corresponding eigenvalues.
Note $^T$ denotes transpose and $\mathcal{R}(\cdot)$ returns the real part of a complex number.