Orthogonal matrix of symmetric matrix (eigenvectors)

Question

I have this exercise:

First of all, I can't find such $v_3'$ such that $v_2 · v_3 = 0$. Second, how is the orthogonal matrix $P$ when the normalized vectors $v_1, v_2$ and $v_3'$ have different lengths?

Answer 1

Hint: since $v_2$ and $v_3$ live in the same eigenspace (for $\lambda=-3$) which is of dimension$~2$, you can replace $v_3$ by some linear combination$~v_3'$ of $v_2$ and $v_3$ (with nonzero coefficient of $v_3$) which will still be an eigenvector in that eigenspace. This allows you to make $v_3'$ orthogonal to $v_2$. Then divide all basis vectors by their norm.