Submatrices with the best-bounded inverses: revisiting the hypothesis

03/13/2023

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by   Yuri Nesterenko, et al.

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The following hypothesis was put forward by Goreinov, Tyrtyshnikov and Zamarashkin in <cit.>. For arbitrary semi-orthogonal n × k matrix a sufficiently "good" k × k submatrix exists. "Good" in the sense of having a bounded spectral norm of its inverse. The hypothesis says that for arbitrary k = 1, …, n-1 the sharp upper bound is √(n). Supported by numerical experiments, the problem remains open for all non-trivial cases (1 < k < n-1). In this paper, we will give the proof for the simplest of them (n = 4, k = 2).