Suppose there is a non-normal block matrix like following: \begin{equation} P_{1}= \left[\begin{array}{cc} 0 & I\\ A_{1} & A_{2}\\ \end{array}\right] \end{equation} in which $0,I$ are respectively zero and unit matrices, $A_{1},A_{2}$ are $n*n$ real matrix. And $P_{1}^{T}P_{1} \ne P_{1}P_{1}^{T}$, which means that $P_{1}$ is a non-normal matrices. Though $P_{1}$ cannot be orthogonal diagonalized, there may exist some non-singular(non-unitary) matrices to diagonalize it.There is a example Can non-normal matrices with double eigenvalues never be diagonalized?.
Now I am wondering in what conditions can a non-normal matrix $P_{1}$ be diagonalized. To be more specific, what conditions should matrices $A_{1},A_{2}$ satisfy to make $P_1$ diagonalizable?
I have got some knowledges that if $P_1$ has $2n$ linearly independent eigenvectors, then $P_1$ is similar to a diagonal matrix. However it is not easy to calculate the eigenvectors and dimension of eigenvectors space, so i want to get some more verifiable conditions represented by $A_1,A_2$, Thank you very much!