Suppose we have a rotation matrix $P$ and a diagonal matrix $D$. How do you explain the effect of $D$ as a linear transformation on the plane $\mathbb{R}^2$ geometrically? What about the effect of $P^{-1}DP$?
My approach: Well whenever we apply a diagonal matrix \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} on a vector, we just rescale the vector thus we just stretch the $\mathbb{R}^2$ plane.
Then for $P^{-1}DP$ we first rotate the vector, stretch and then bring it all back. So in this case what is the answer?
The effect of $D$ is a rescaling with dilation/contraction of the $2$ axes.
The effect of $P^{-1}DP$ is a rescaling with dilation/contraction in the direction $-\theta$ and $-\theta + 90°$.