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Nov 3, 2015 at 13:29 comment added Jeff the question is updated.
Nov 3, 2015 at 13:11 comment added Igor Rivin @Hossein But for any $n$ the probability is $0,$ so the limit is $0.$ As for "almost unitary", you have to define what that means.
Nov 3, 2015 at 12:57 comment added Jeff As $n\to \infty$, the number of those $r \times r$ submatrices goes to infinity as well. Right? In any case, I modified the question to cover ``almost unitary'' matrices. Do you have any idea about that?
Nov 3, 2015 at 12:47 history answered Igor Rivin CC BY-SA 3.0