Characterization of Diagonizability

Theorem

$T:V\to V$ Diagonalizable $⟺$ if $[T{]}_{\alpha }^{\alpha }$ is similar to the diagonal matrix

Proof

Proving $⟹$

• If $T$ is diagonizable, we have a basis $a=\{v_1,\dots ,v_n\}$ such that $T(v_i)={\lambda }_i{v}_i$ for eigenvalues ${\lambda }_i$
• If we have $[T{]}_{\alpha }^{\alpha }$ in this basis, we get:

$$[T{]}_{\alpha }^{\alpha }=\left[\begin{array}{c}{\lambda }_1\dots 0\\ 0\dots {\lambda }_n\end{array}\right]$$

• $=ID{I}^{-1}$. Therefore, $[T{]}_{\alpha }^{\alpha }$ is similar to a diagonal matrix

Proving $⟸$