Set of Rotation Matrices in $R^n$ Which Fix (1,1,…,1)

Question

For the case where $n=3$, I was given these rotation matrices which fix the vector $(1,1,1)$.

How would I generalize these matrices for $R^n$?

Answer 1

Select any two length-1 and orthonormal unit (column-) vectors $u$ and $v$ such that $u$ and $v$ are both orthogonal to $(1,1,\dots,1)$. For any angle $\theta$, the matrix $$ M = I + \pmatrix{u & v}\pmatrix{(\cos \theta - 1) & -\sin \theta\\ \sin \theta & (\cos \theta - 1)}\pmatrix{u & v}^T\\ = I + (\cos \theta - 1)(uu^T + vv^T) + \sin \theta(vu^T - uv^T) $$ represents a rotation within $n$-dimensional space that fixes the vector $(1,1,\dots,1)$. In particular, this rotation occurs within the plane spanned by $u$ and $v$, which importantly is orthogonal to $(1,1,\dots,1)$.

Keep in mind, however, that for "rotations" in $n$-dimensional space with $n \geq 4$, it is possible to have multiple simultaneous rotations in mutually orthogonal planes. This transformation only applies a transformation in a single 2-d plane.