Why diagonal matrix doesn't change eigenvalues?

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! :)

Re: Why diagonal matrix doesn't change eigenvalues?

If I am understanding you correctly, this is **not** true.

For example, suppose $\displaystyle A= \begin{bmatrix}-1 & 2 \\ -6 & 6\end{bmatrix}$. It is easy to see that A has eigenvalues 2 and 3.

Now, take $\displaystyle D= \begin{bmatrix}\frac{3}{2} & 0 \\ 0 & 2\end{bmatrix}$, a diagonal matrix with determinant 1.

$\displaystyle DA= \begin{bmatrix}-\frac{3}{2} & 2 \\ -12 & 12\end{bmatrix}$

which does NOT have 2 and 3 as eigenvalues.

Re: Why diagonal matrix doesn't change eigenvalues?

hmm...This is weird. Earlier on, I tried having several matrices having my $\displaystyle D=\begin{bmatrix}2 & 0\\ 0 & \frac{1}{2}\end{bmatrix}$ and then many different matrices for $\displaystyle A$ such as $\displaystyle A=\begin{bmatrix}1 & 3\\ 4 & 2\end{bmatrix}$, $\displaystyle A=\begin{bmatrix}1 & 14\\ 8 & 2\end{bmatrix}$, $\displaystyle A=\begin{bmatrix}1 & 2\\ 3 & 2\end{bmatrix}$, etc etc, they all worked.

After your example, I tried more values into the matrices and began to realise that it doesn't work all the time. I'm curious if there is any kind of special properties that allow this proposition to be true? (Wondering)