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
What's written here? Is it ?
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).
No, this isn't correct: not only but ALSO they both have the same irreducible factors! Thus, must be divided by each and all of the following factors: , and this already makes our work much easier.
As every root (or irreducible polynomial) of C(x) is also a root of M(x),
And this is exactly what I meant above...
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?
Yes. Google "companion matrix". For example, let us build a matrix whose minimal pol. (and also characteristic, of course...these companion matrices kick ass!) is
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"....