Essentially, I am running an algorithm where I create some matrix G. When I work G on small systems it is not apparent that there is any pattern within G. However, when G becomes sufficiently large it is evident that it was developing into a strong block matrix structure.
Now, I was thinking: "Is there some way I can use this interesting property of G in order to speed up its inversion process? Especially when the dimensions of G get quite considerable" However I am at a loss, since there really isn't a pattern local to each block matrix which repeats across the entire matrix structure. Rather, I just see that block patterns are indeed emerging in general, and yes, they look locally similar next to one another, but I believe unless each block has a nice, consistent, persistent, internal property like they are: "all exactly the same" or "they are all diagonal", it is "technically" not that different to any other matrix inversion problem.
However, it scorches my brain and I'd like ti use this information somehow, but currently have no idea. Because there aren't any repeating patterns between the blocks, because then I was considering ---> If there is no pattern that is usable, why don't we in practice see a random 100 by 100 matrix, and split it into 10 by 10 blocks and invert those smaller blocks (so instead of one large 100 by 100 matrix inversion, there would be 100, 10 by 10 matrix inversions (and a lot of internal multiplications) ... and is there any reason to believe that is faster in the slightest?).
As you can see a nice structure emerges and would love hear if anyone has any ideas. For a bit more information, these matrices were constructed by following a combinatorial formula, but I don't think I'm privy to say which one.
I will attach images below:
When G has 30 by 30 blocks:
When G has 10 by 10 blocks:
When G is 5 has 5 blocks:
When G is 4 by 4 "blocks" but it cant be seen apparently:
