I just finished reading some basics on the Jordan normal form ( mostly just an existence proof for functions where the characteristic polynomial splits).
I determined the JNF for some basic matrices/functions without running into any problems. The cases I ran into where of the sort where the geometric and algebraic multiplicities of the eigenvalues where for example 1 and 3.
In this case I know that the desired basis must consist of of one cycle. This gives me some basic equations which I can solve to find possible endpoints for the desired cycle.
Now I am thinking what I should do if I encounter other cases, for example where the algebraic multiplicity is 6 and the geometric mult. is 3. How would I efficiently solve this? Should I first try to find a cycle of length 4? Does such a cycle even necessarily exist?