I am just here to understand a basic concept of linear algebra.
Let be v an endomorphism and its matrix in the canonical basis is given by $A = \begin{pmatrix} 1&5&6&3 \\ 1&3&4&5\\4&3&7&8 \\ 1&7&4&3 \end{pmatrix}$, which is just an example (I'm not going to do the calculus because this is not important here). So, it's not difficult to find a characteristic polynomial and another basis $B = (v_1,v_2,v_3,v_4)$, in which I can triangulate A.
But now, I have another basis, which is cyclic. Let's call it $B' = (w,v(w),v^2(w),v^3(w))$, with $w = v_1 + v_3$. Again, this is not complicated to do.
However, how do I find the matrix of v in this particular basis using the characteristic polynomial?
Thanks for your help!