Let's first proof T is diagonalizable V = E1 + E2.
If T is diagonalizable, it can be written as , where D is a diagonal matrix with its eigenvalues on the main diagonal, and P represents a basis of eigenvectors.
That basis spans V.
Therefore V is the direct sum of the eigen spaces.
Now let's proof T is diagonalizable.
This means that that basis of together with the basis of span V.
Put those bases next to each other in a matrix P, and put the eigenvalues diagonally in a corresponding matrix D, and you have that
So T is diagonalizable.
Therefore T is diagonalizable iff