I have a quadratic equation of the form: $x_1^2 + 4x_2^2+4x_3^2 + 4x_1x_2 - 4 x_1x_2 - 4x_1x_3 + 2x_2x_3$
My task is to calculate the orthonormal basis for it.
The matrix I made for it: $$ \begin{matrix} 1 & 2 & 2 \\ 0 & 4 & 1 \\ 0 & 0 & 4 \\ \end{matrix} $$
It has two eigenvalues: $e_1$ = 1 and $e_2$ = 4 (of multiplicity two)
I got only two corresponding eigenvectors (1, 0, 0) and (2, 3, 0), respectively, and then got stuck, because if I understand correctly, it is impossible to get a 2x2 basis for a 3x3 matrix.
What am I missing? I guess I should have two eigenvectors for the eigenvalue of 4, but I triple checked and I am still getting only one.
Thanks for the help!