Ok

So i am understanding that any M matrix can be rewritten such that M= UDU^-1, where U consists of the matrix consisting of the eigenvectors and D is the diagonal matrix consisting of eigenvalues.

Is there another way to think about this idea through transformations?

i.e

1. How does the matrix U relate to the matrix M in terms of transformations? Are they related in any way?

2. The matrix D is clearly the enlargement with the scale factors of eigenvalues in x & y directions?.So is it possible to think of the matrix M as a series of three transformation and can someone give me an example where all this makes sense?

Fundamentally, i want to understand why a matrix can be broken up in this way?

Also, and other than the idea of powers is there another application of this result?