I have two random matrices $A$ and $B$, with $N$ columns each. The columns of $A$ and $B$ are independent but not necessarily identically distributed.
$A$ and $B$ may be considered as two instances of an underlying matrix, for example, they may represent two sets of data-driven basis functions learned from datasets of subjects $A$ and $B$.
With this setup, it follows that $M < N$ columns of $A$ have corresponding columns in $B$. That is, the inner product $A_i \cdot B_j$ of columns $A_i$ and $B_j$ is close to $1$ for $M < N$ pairs $(i,j)$, where $M$ is unknown a priori.
The above intuition suggests that all entries of the matrix $G$, where $G_{i,j} = A_i \cdot B_j$ , are not independent of each other.
My question:
if one were to model $A$ and $B$ as random orthogonal matrices, how many degrees of freedom would $G$ have? How does one go about addressing this problem? Analytically? By simulation? Can we derive bounds on this number?
I don't have formal training in mathematical statistics, so any pointers for where to look would be appreciated.
Thanks!