Pardon for a poor title but I have this question which I am very confused:
Let $C$ be a $4 \times 4$ matrix with all eigenvalues $λ = 2, −1$ and following eigenspaces: $E_2=\begin{pmatrix} 1\\1 \\ 1\\1 \end{pmatrix}$ and $E_{-1}=\left(\begin{pmatrix} 1\\2\\ 1\\1 \end{pmatrix},\begin{pmatrix} 1\\1 \\ 1\\2 \end{pmatrix} \right)$. Calculate the $C^4u $ for $u=\left[\begin{pmatrix} 6\\8 \\ 6\\9 \end{pmatrix}\right]$ if possible, and explain if not possible.
Now my understanding was to first get the diagonal matrix of $C$ so that we can easily find the powers of C. We already have eigenvectors from the basis of eigenspaces so we can get the coordinate matrix $Q$ and we can write $D=Q^{-1}CQ$, where D is diagonal matrix consisting of eigen values.
Problems: Firstly I cant seem to prove that C is diagonalisable or not, then only we can write the above equation. For that I need to check whether the characterestic polynomial splits or not (which might as we got the eigenvalues), and then check muliplicity of the eigenvalues and verify if $\text{multiplicity}(\lambda)=\text{dim}(E_\lambda)$, and we dont have muluplicity given, so I am kinda stuck. I also need multiplicity to write the diagonal matrix.
So I need help here, as I am stuck and cant seem to go other way. Thanks in advance.