How to find the Jordan Canonical Form of a 5x5 matrix and its steps?

[TMO]

To see the steps I have completed so far, https://math.stackexchange.com/q/3168898/261956

I think there are at least three more steps. The next step is finding the eigenvectors together with the generalized eigenvectors of each eigenvalue. Then we use this to construct the transition matrix. Then we do matrix multiplication in order to

 

[fresh_42]

If you already have the eigenvalues of

\begin{pmatrix}177& 548& 271& -548& -356\\ 19& 63& 14& -79& -23\\ 8& 24& 17& -20& -20\\ 42& 132& 55& -141& -76\\ 56& 176& 80& -184& -105\end{pmatrix}

with their multiplicity, then I read from your table
\begin{array}{c|c|c}
\lambda & \operatorname{am}_C(\lambda) & \operatorname{gm}_C(\lambda) \\ \hline
3 & 4 & 2 \\
-1 & 1 & 1
\end{array}
the Jordan normal form $\begin{pmatrix}3&1&0&0&0\\0&3&0&0&0\\0&0&3&1&0\\0&0&0&3&0\\0&0&0&0&-1\end{pmatrix}$

Given the informations are correct, what do you want to know?