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.


(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.

  • $\begingroup$ 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 Nov 3 '15 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 Nov 3 '15 at 13:11
  • $\begingroup$ the question is updated. $\endgroup$ – Jeff Nov 3 '15 at 13:29

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.