Approximate diagonalization (eigendecomposition) of a non-symmetric matrix

Question

Suppose that $A\in\mathbb R^{n\times n}$ is a non-symmetric and diagonalizable matrix with $A=PVP^{-1}$, where $P,V\in\mathbb R^{n\times n}$ and $V$ is a diagonal matrix. Here I have only an estimate $\hat A$ of $A$ which was obtained by statistics. Is there any way to approximately decompose $\hat A$ as $\hat A\approx \hat P\hat V{\hat P}^{-1}$ with real-valued $\hat P$ and real-valued diagonal $\hat V$? ($\hat A$ may not be diagonalizable).