Orthogonal Basis of Zero-Sum Space

Question

I would like to find an orthogonal basis of the zero-sum space $\mathcal{V}^{(N)}_0$, defined as the space of all $N$-dimensional real vectors $v$ such that $\sum_i v_i=0$.

That would mean finding a set of $N -1$ vectors that are both orthogonal to each other and orthogonal to $v_0=(1,1,\ldots,1)$.

I tried choosing $v_2=(-1,1,0,\ldots,0),v_3=(-1,0,1,\ldots, 0),\ldots,v_n=(-1,0,0, \ldots, 1)$. These are all orthogonal to $v_0$, but not to each other, because for instance $\langle v_2,v_3\rangle=1$.

Answer 1

You can apply Gram-Schmidt to your choice of vectors to obtain the orthogonal basis.

Answer 2

You can just apply the Gram-Schmidt algorithm to your basis to force it to be orthogonal. Alternatively, a variation on your basis would be $v_n=(\underbrace{1,\ldots, 1}_{n-1} ,1-n,0, \ldots, 0)$, so $v_2=(1,-1,0,\ldots, 0)$, $v_3=(1,1,-2,0,\ldots, 0)$ and so on. This basis is orthogonal by virtue of the fact that for $i\neq j$, $(v_i,v_j)$ is the sum of components of $v_{\min(i,j)}$ which is always $0$.