Question

Please argument all your answers and explain your arguments so i can understand better dont use advanced things im just taking linear algebra course.

Let V be a vector space of finite dimension over $\mathbb{C}$.

$T_1,T_2$ linear operators over V that conmute. Show that $T_1$ and $T_2$  have at least one common eigenvector

Answer 1

Given T_1: v rightarrow v T_2 v rightarrow v such that T_1 T_2 = T_2 T_2. Let lambda is eigen value of T_1 x be eigen vector of T_1 corresponding to eigen value Therefore T_1 (nu) lambda x Then we claim That the vector v = T_2 (nu) belongs to the eigen space F_lambda of lambda we have T_1 T_2 = T_2 T_1 Rightarrow T_1 (nu) = T_1 T_2 nu = T_2 T_1 nu = lambda T_2 nu = lambda nu Hence nu elementof E_x ie T_1 (nu) elementof E_lambda T_1 & T_2 have at least one common eigen vector