# How to determine the degrees of freedom of an inner-product matrix of two random matrices?

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, i.e., the inner product {A_i, B_j} of columns A_i and B_j is close to 1 for M < N pairs (i,j). M in unknown a priori.

The above intuition suggests that all entries of the matrix G, where G(i,j) = {A_i, B_j}, are not independent of each other.

My question is: 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!

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