Suppose I have this: $\displaystyle A=P_{0}\Lambda P_{0}^{-1}$

And then there is a diagonal matrix $\displaystyle D$ such that its determinant is always equals to 1: $\displaystyle det(D)=1$

Then, for some reason, $\displaystyle DA = P \Lambda P^{-1}$ is always true.

How can I show that for the lambda, which is the eigenvalues matrix, does not change for all $\displaystyle DA$?

And what is the relationship between $\displaystyle P$ and $\displaystyle P_{0}$? I understand that their determinant $\displaystyle det(P_{0}\Lambda P_{0}^{-1})=det(P \Lambda P^{-1})$ but still, what's the relationship between the $\displaystyle P_{0}$ and $\displaystyle P$ because $\displaystyle P_{0}$ is eigenvectors matrix for $\displaystyle A$ while $\displaystyle P$ is eigenvectors matrix for $\displaystyle DA$ and so they are different. But still, there isn't any strong relationship, is there?

Thanks!