I want to find the asympototic behaviors of $||M^nv||$ with some rate. [Just claiming that it's divergent is not enough - it requires finding some parallel bounds.]
We are given that M is 2x2 with characteristic polynomial $t^2-6t+9$.
(1) How does the asymptotic behavior of $||M^nv||$ have anything to do with diagonalizability of a matrix?
(2) I only saw that it simulates asymptotically to $3^nv$, but I was told that there is another situation involving $n-1$ somewhere. Is this the situation when it's diagonalizable or not? What is the other situation and how is it derived?
This is not a homework question and I don't really know which topic this problem belongs to. Any help with two questions above will be appreciated!
You can have two slightly different cases, as you suspected.
In the diagonalizable case, in some basis you get $M=\pmatrix{3&0\\0&3}$ so that indeed $M^nv=3^nv$.
In the non-diagonalizable case, there is a basis where you get the Jacobi normal form $M=\pmatrix{3&1\\0&3}$ so that $M^n=3^n\pmatrix{1&\frac{n}{3}\\0&1}$. So indeed, the norm (in the original basis) of the matrix-power-vector product can have components that behave like $n3^{n-1}$.