Unusual definition of the sphere $S^n$

Question

In a paper I’m reading, the author defines the sphere $S^{m(k-1)-1}$ ($k \geq 2$) as the set of $m \times k$ matrices $(a_{ij})$, with $a_{ij} \in \mathbb{R}$ and satisfying the following two properties:

$$\forall i, \sum_{j=1}^k a_{ij} = 0$$ $$\sum_{i,j} a_{ij}^2 = 1$$

Of course the usual definition is $S^n = \{x \in \mathbb{R}^{n+1} | \|x\|=1\}$.

Equivalence for $k=2$ is pretty easy: $a_{i1}^2 + a_{i2}^2 = 2a_{i1}^2$, since $a_{i2}=-a_{i1}$, so you just multiply the left column by $1/\sqrt{2}$ to get the usual mapping. I’m having trouble seeing how this parameterization works for larger $k$.

Also, for some context, I think the reason the author wants to do this is so that shifting columns is some desired $k$-cycle of points.

Answer 1

First notice that $(a_{ij})_{ij} \mapsto \sum_{i, j}a_{ij}^2$ is a positive definite quadratic form $q$ on the $\Bbb R$-vector space of all $m \times k$ matrices.

Next, the matrices which satisfy $\sum_j a_{ij} = 0$ form a subvector space $V$ of dimension $m(k - 1)$.

Thus $q$ induces a positive definite quadratic form on $V$, and hence the subset $\{v \in V: q(v) = 1\}$ is a sphere of dimension $m(k - 1) - 1$, up to choosing a basis of $V$.