Suppose I have this:

And then there is a diagonal matrix such that its determinant is always equals to 1:

Then, for some reason, is always true.

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

And what is the relationship between and ? I understand that their determinant but still, what's the relationship between the and because is eigenvectors matrix for while is eigenvectors matrix for and so they are different. But still, there isn't any strong relationship, is there?

Thanks!