The following matrix is given:
After doing all the usual calculations I got the following polynomial and eigenvalues
Now we know that for the eigenvalue of 1, the algebraic multiplicity is going to be 2 (due to the power of two in the polynomial), and we know the algebraic multiplicity for I is going to be 1, and for -i it is also going to be one.
I also found that the geometric multiplicity for the eigenvalue of 1 is going to be 1 (found one eigenvector).
Lastly, the question asks to give the Jordan Normal form for this matrix. Now I know that for the Jordan form, we will have the eigenvalues on the diagonal and 1s or 0s everywhere else. Then we are going to have different blocs for each eigenvalue depending on its algebraic multiplicity. So here we know that we are going to have two 1s on the diagonal as well as i and -i. Since the geometric multiplicity of the first eigenvalue is 1, that means that we are going to have both of these 1s in one bloc with another 1 above the diagonal.
Now my question is how are these blocs supposed to be ordered in the diagonal? Is there a specific order that should be followed, and why?
From what I know all the following solutions should be correct
However, the answer to the exercise states that there is a unique solution and it is this one:
Why is this the only form in this case?
And in other cases with blocs of different algebraic and geometric multiplicities is there a specific rule that should be followed for the order of the blocks?
Your interpretation is right. Unless the book prescribes a recipe for it, there is no canonical way of ordering the blocks in the Jordan form. And an actual recipe would have to be fairly complicated, because not only you would have to prescribe an order between the different (complex) eigenvalues, but also between the blocks of different size corresponding to the same eigenvalue.