Hello, everyone!

Supposing that there is a unit vector in $n$-dimensional real space $\mathbf{x}_1\in\mathbb{R}^n$, I want to get a group of $n-1$ vectors to form an orthogonal basis with $\mathbf{x}_1$. One way to achieve this goal is to firstly randomly generate $n-1$ linear independent vectors and then orthogonalize them use the Gram-Schmidt process to obtain the orthogonality.

Then, I wonder if it is possible to achieve this goal by a group of orthogonal transformations, i.e., if there are $n-1$ orthogonal matrices $\{A_i\}\_{i=2}^n$ so that $\mathbf{x}_i^\top\mathbf{x}_j=0,\forall i\neq j$ where $\mathbf{x}_i=A_i\mathbf{x}_1,i=2,\ldots,n$.

Is there any result about this question, please? Any suggestion will be welcome. Thank you very much!

## Comment

In the special case where $n=2$, it is possible by $A_2=\begin{bmatrix} 0 & 1 \\\ -1 & 0 \end{bmatrix}$. It is straightforward to check that the orghotonality of $A_2$ and $(A_2\mathbf{x}_1)^\top\mathbf{x}_1=0,\forall\mathbf{x}_1$. It is easy to comprehend because $A_2$ is indeed a ration of 90 degree in a 2-dimensional plane.

However, it is not so simple in the case of $n=3$. First, to obatin the orthogonality by a transformation $A_2$, i.e., $\mathbf{x}^\top A\_2\mathbf{x}=0,\forall\mathbf{x}\in\mathbb{R}^n$, the following equation need to be satisfied. \begin{align} &a\_{11}=a\_{22}=a\_{33}=0\\\ &a\_{12}+a\_{21}=a\_{13}+a\_{31}=a\_{23}+a\_{32}=0 \end{align} where $a_\{ij}$ is the $i,j$-th element of $A_2$.

With the conditions above, it is straightforward to check that $|A_2|=0$ and thus $A_2$ cannot be an orthogonal matrix, which means there does not exist an orthogonal transformation to obtain an orthogonal vector to a given one.

In the case of $n=4$, it is possible to find an orthogonal transformation to obatin an orthogonal vector to a given one. But I did not find how to find 3 such transformations to form an orthogonal basis including the given vector.