We start with a diagonal matrix:
Here is a Quicktime animation (alternate version) that illustrates how a diagonal matrix transforms a square.
We form a matrix A that is similar to D using the matrix P.
Note that A has the same eigenvalues as D.
Therefore, A transforms area by the same factor as D.
However, it looks as if A transforms the plane in a more complicated fashion than D does.
But if we use the right choice of coordinates, A transforms the plane (alternate version) in the "same" way as D.