Find a set of orthogonal vectors in $\mathbb{R}^{n}$ dimension with their components summing to zero

Question

I am interested in finding a set of, say 3, orthogonal vectors whose components add up to zero. This is a concept that is used in statistics as well, but for the sake of this question, I wanted to phrase it as simple as I could.

I have found a built-in function within my statistical software to do this task for me, but I do not want to blindly use it without understanding the "behind the scene" action.

I also have found a notion in my linear algebra course discussing about the Orthogonal Complement of a span, but the components of these vectors do not have to add up to zero.

Any tip would be appreciated!

Answer 1

The set of vectors with coordinates summing to zero are the linear span of the vectors $e_1-e_2$, $e_2-e_3,\ldots,e_{n-1}-e_n$ where the $e_i$ are the usual unit vectors.

You can use the Gram-Schmidt process on these to produce up to $n-1$ orthogonal vectors.

Answer 2

There is no general solution in $\mathbb{R}^n$, but there are solutions in $\mathbb{C}^n$. You can convince yourself by trying to solve the following equations, where $a = [a_1, a_2], b = [b_1, b_2]$ are vectors in $\mathbb{R}^2$:

$$a_1b_1 + a_2b_2 = 0$$ $$a_1 + b_1 = 0$$ $$a_2 + b_2 = 0$$

Solving the equations with WolframAlpha gives solutions in $\mathbb{C}^2$ only.