Given a minimal polynomial, find a transformation
Could you help me with the last part of this question?:
Let V be a vector space over C, and let T:V->V be a linear transforamtion with characteristic polynomial C(x)=(x^2+1)(x-1)^2\x^2 How many possibilities are there for the minimal polynomial? For each possibility, give an example of a suitable linear transformation.
Now, as the minimal polynomial (M(x)) divides the Characteristic polynomial, then M(x) is either x+i, x-i, x+1, 0, or any combination of these (sometimes up to powers of 2).
As every root (or irreducible polynomial) of C(x) is also a root of M(x), then we are left with four possibilities (where \ means we're no longer in the superscript):
M1=(x^2+1)(x-1)x , M2=(x^2+1)(x-1)^2\x, M3= (x^2+1)(x-1)x^2, or M4=(x^2+1)(x-1)^2\x^2
What I'm struggling with is finding functions whose minimal polynomials are M1, M2, M3, and M4, respectively. Is there an algorithm for this?
I know that the degree of the characteristic polynomial equals the dimention of the vector space. Hence, dim(V)=6. I'm thinking about representing the functions (F1-F4, respectively) as diagonal or upper triangular matrices (as otherwise the calculations would be enormous), but I'm not sure where to start from. (matrices wrt the standard basis) A friend suggested using the Jordan normal form, as well (almoust-diagonal matrices).
Could anyone offer any suggestions/do one as an example?
Your help is greatly appreciated!
P.S. Sorry for the format, but I'm new to LaTex, and all formulas I tried ended up as "syntax errors"....