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Let $\mathbf{R} \in \mathbb{C}^{~m \times n} $ with $m \leq n $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Let where $r$ be the rank of $\mathbf{R}$. Three questions:

(1) can we say $r = \min(m,n)$?

(2) what is the probability of having a unitary submatrix of size $r$ inside $\mathrm{R}$?

(3) if the answer of question 2 is not available in general, is there any asymptotic result? For instance, existence of at least one unitary submatrix of size $r$ almost surely as $n\to \infty$?

PS: what about ``almost unitary'' submatrices for questions (2) and (3)?A definition for almost unitary matrix can be that, $\mathbf{U}^+ = \mathbf{U}^H + \mathbf{e}$, where $\mathbf{e}$ is a small error matrix (small entries), and $+$ is the Moore–Penrose pseudoinverse operation.

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(1) a.s yes (2) The probability is $0.$ Unitary submatrices are of positive codimension in $GL(r),$ and since there only a finite number of $r\times r$ submatrices, the probability is $0.$ If you want "almost" unitary", there might be something one can say. (3) No, the probability is always 0.

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  • $\begingroup$ 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? $\endgroup$
    – Jeff
    Commented Nov 3, 2015 at 12:57
  • $\begingroup$ @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. $\endgroup$
    – Igor Rivin
    Commented Nov 3, 2015 at 13:11
  • $\begingroup$ the question is updated. $\endgroup$
    – Jeff
    Commented Nov 3, 2015 at 13:29

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