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Off the top of my head, without a real answer, I would think that if you looked up something like Singular Value Decomposition, you might find something that essentially answers this question. What I'm about to say could be very wrong, but :
if A = LDU (or something like that) where L and U are a set of matrices to permutations to orthogonalize A leaving behind only a diagonal matrix D, then if you take the inverse of A
A_inv=inv(LDU) =? U_invD_invL_inv (maybe the ordering doesn't switch for inverses...I can't remember) then you can see that D_inv will be a matrix which contains the values of (1/d_i) where d_i were the diagonal entries of the original matrix D. Assuming that D isn't singular, then something along these lines should point towards A_inv being orthogonalized diagonally.
I think. At least it might be food for thought
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