I have the problem
Let T be the linear operator on $\mathbb{C}^3$ which is represented in the standard ordered
basis by the matrix $\left( \begin{array}{ccc}
1 & i & 0 \\
-1 & 2 & -i \\
0 & 1 & 1 \end{array} \right)$ .Find the T-annihilator of the vector (1, 0, 0) . Find the T-annihilator of (1, 0, i) .
I have for a definition that the T-annihilator of a vector is $P_\alpha$ the unique monic polynomial which generates the ideal such that $g(T)\alpha = 0$ for g in this ideal.
However I don't see how I'm supposed to find this generator. I'm pretty sure it has to deal with the minimal polynomial of T but I'm stuck after that.
Can anybody point me in the right direction?
Best Answer
Show that the $T$-annihilator divides the minimal polynomial. From that point on, just test the factors of the minimal polynomial to find the one of smallest degree which annihilates the vector.