My variables are: $O_1,\dots,O_M\in \mathbb{O}(d)$
I have equations like:
$O_{1_1} x_{1,1}+ \dots + O_{1_{m_1}} x_{1,m_1} = x_{1,(m_1+1)}$
$\vdots$
$O_{n_1} x_{n,1}+ \dots + O_{n_{m_n}} x_{n,m_n} = x_{n,(m_n+1)}$
$i_j \in [1,M]$ and $x_{i,j}\in \mathbb{R}^d$ are constants.
My question is:
What is the value of $n$ such that the above equations have a unique solution?
My guess: $n$ is related to the dimension of $\mathbb{O}(d)$.