Orthonormal basis of $\{(x_1, \dots, x_n) \mid x_1+x_2+\cdots+x_n=0\}$

Question

How can we find the orthonormal basis of $\{(x_1,\dots,x_n) \in \Bbb R^n \mid x_1+x_2+\cdots+x_n=0\}$?

It is easy to find a basis, but using Gram-Schmidt procedure seems difficult to obtain an orthonormal one.

Answer 1

You need to find a orthonormal (sub-)basis of $n-1$ elements in ${\Bbb R}^n$ that is orthogonal to $v=(1,1,\ldots,1)$. I don't think there is a good canonical choice. Here is a simple way, though: \begin{align} u_1 &= \left( \begin{matrix}1 & -1 & 0 & 0 & \ldots & 0 \end{matrix} \right)\\ u_2 &=\left( \begin{matrix}1 & 1 & -2 & 0 & \ldots & 0 \end{matrix}\right) \\ u_3 &=\left( \begin{matrix}1 & 1 & 1 & -3 & \ldots & 0 \end{matrix}\right) \\ & ...\\ u_{n-1} &= \left( \begin{matrix} 1 & 1 & 1 & 1 & \ldots & -(n-1) \end{matrix}\right) \\ \end{align} They are pairwise orthogonal and orthogonal to $v$ and you just have to normalize.