On the Distance of a Large Toeplitz Band Matrix to the Nearest Singular Matrix
Albrecht Böttcher, Sergei M. Grudsky, A. V. Kozak · Birkhäuser Basel eBooks · 2002
If the symbol of an infinite Toeplitz band matrix has nonzero winding number, then the distances of the n × n truncations of this matrix to the nearest singular matrices go to zero with exponential speed as n → ∞. The purpose of this note is to show that if we measure the distance to the nearest singular matrix within the matrices of the same Toeplitz band structure as the original matrix, then things are less dramatic: this distance stays away from zero if only the origin does not belong to the Schmidt-Spitzer set. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.