For k=2 and any n, every n x 2 orthonormal matrix U has a 2 x 2 submatrix Q with smallest singular value at least 1/sqrt(n).
A theory of pseudoskeleton approximations.Linear algebra and its applications, 261(1-3):1–21, 1997
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On the submatrices with the best-bounded inverses
For k=2 and any n, every n x 2 orthonormal matrix U has a 2 x 2 submatrix Q with smallest singular value at least 1/sqrt(n).