Determine an orthonormal basis so that $s(v_i, v_j) = 0, 1 \leq i, j \leq 3, i \not= j$

Question

Determine an orthonormal basis $ (v_1, v_2, v_3) $ so that $ s(v_i, v_j) = 0, 1 \leq i, j \leq 3, i \not= j $

$s$ is a symmetrical bilinear form given by the matrix A:

$$ A = M_\beta(s) = \pmatrix{-3 & 0 & -1 \\ 0 & -3 & 0 \\ -1 & 0 & -1} $$

So how can I find those three vectors?

I calculated the quadratic form $ q(x_1, x_2, x_3) = -3x_1y_1 - 3x_2y_2 - x_3y_3 -x_3y_1 - x_1y_3 $.

Answer 1

First make an arbitrary (nonzero) choice for $v_1=(x_1,x_2,x_3)$, say $x_1=1,x_2=x_3=0$. A vector $(y_1,y_2,y_3)$ is orthogonal to $v_1$ iff $q(1,0,0,y_1,y_2,y_3)=0$, i.e. $y3=-3y_1$. So $v_2$ will be of the form $(y_1,y_2,-3y_1)$, and $v_2$ will be of the form $(z_1,z_2,-3z_1)$. Now we make an arbitrary (nonzero) choice for $y_1$ and $y_2$ : say $y_1=1$ and $y_2=0$. This gives $v_2=(1,0,-3)$. And $v_3$ must be orthogonal to $v_2$, so that $q(1,0,-3,z_1,z_2,-3z_1)=0$, i.e. $z_1=0$. So $v_2=(0,z_2,0)$, and we make the arbitrary choice $v_2=(0,1,0)$.