Orthogonal diagnoal Matrix

Question

I'm having problem for orthogonal diagonalization a matrix.

A=\begin{pmatrix} 1 & 1 & 0\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}

I want to know what I'm doing wrong. So in order to find this kind of matrix I need to get to this step: $P^{-1} A P$

Now, I need to find the characteristic polynomial which is $p(x) = -x^3 + 3x - 2x = 0$

therefore $x = 2, 1, 0$

What is the next step?

Answer 1

Steps:

1) Find an eigenvector for each of those three eigenvalues

2) Apply Gram-Schmidt to the above three vectors (why is it possible? Say, why are they linearly independent?)

3) Now form $\;P\;$ with the orthonormalized basis as columns

4) You're done.