Diagonalizable Properties Theorem

Theorem

1. Let $T\in \mathcal{L}(V)$, $dim(V)<\infty$
2. Suppose $T$ is diagonalizable and $c_1,\dots ,c_k$ are distinct Eigenvalues of $T$
3. There $\exists E_1,\dots ,E_k\in \mathcal{L}(V)$ such that:
   1. $T=c_1{E}_1+\cdots +c_k{E}_k$
   2. $I=E_1+\cdots +E_k$
   3. $E_i{E}_j=0,\forall i\ne j$
   4. $E{j}^2=E_j,\forall j$
   5. $range(E_i)=ker(T-e_{i}I)$

Proof

1. Suppose $T$ is diagonalizable with distinct Eigenvalues $e_1,\dots ,e_k$.
2. let $W_i$ be the space of characteristic vectors associated with characteristic values $c_i$
3. As we have that $\forall e,W_1\oplus \cdots \oplus W_k$
4. Let $E_i$ be the projections associated with the decomposion as with property 1, Then, the last 4 conditions are satisfied

Proof of Property 1

1. We have shown that $\alpha =E{\alpha }_1+\cdots +E{\alpha }_k$
2. Then, $T\alpha =TE{\alpha }_1+\cdots +TE{\alpha }_k$
3. That is to say, $T=c_1{E}_1+\cdots +c_k{E}_k$