Math Help - Show that T is diagonalizable

1. Show that T is diagonalizable

Suppose T: V----> V has only two distinct eigenvalues k1 and k2

Show that T is diagonalizable iff V = Ek1+ Ek2

2. Re: Show that T is diagonalizable

Originally Posted by dave52
Suppose T: V----> V has only two distinct eigenvalues 1 and 2

Show that T is diagonalizable iff V = E1 + E2
Hi dave52!

It seems to me that it is not true.

$\begin{bmatrix}1&0&0 \\ 0&1&0 \\ 0&0&2\end{bmatrix}$

This matrix is diagonal, has distinct eigenvalues 1 and 2, but $\mathbb R^3 \ne E_1 + E_2$.

3. Re: Show that T is diagonalizable

I think you meant to say that your vector space has Dimension 2.

4. Re: Show that T is diagonalizable

sorry I just fixed the problem

5. Re: Show that T is diagonalizable

Let's first proof T is diagonalizable $\Rightarrow$ V = E1 + E2.

If T is diagonalizable, it can be written as $PDP^{-1}$, 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 $V = E_1 + E_2$ $\Rightarrow$ T is diagonalizable.

This means that that basis of $E_1$ together with the basis of $E_2$ 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

$TP = PD$

$T = PDP^{-1}$

So T is diagonalizable.

Therefore T is diagonalizable iff $V = E_1+ E_2 \qquad \blacksquare$