From this video, I learned that applying the transformation $T$ to a vector $v$ $n$ times is amount to n times of transforming the vector into eigenbasis, applying the diagonalized matrix, and then converting back again. And this conversion makes the calculation much easier.
The trick is simple and just like that in Doing a transformation in a changed basis, but what confuses me is why the $D$ and $C$ depicted below happen to be composed of $T$'s eigenvalues and eigenvectors?
Can $D$ and $C$ be composed of other values/vectors? Can the diagonal values of $D$ be eigenvalues but $C$ not eigen vectors? Or $C$ be eigen vectors but $D$ be non-eigenvalues?
How to prove that it's not a coincidence?