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Comments on the approximate factorisation of matrix functions with unstable sets of partial indices

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WHTW01 - Factorisation of matrix functions: New techniques and applications

It is well known for more than 60 years that the set of partial indices of a non-singular matrix function may be unstable under small perturbations of the matrix [1]. This happens when the difference between the largest and the smallest indices is larger than unity. Although the total index of the matrix preserves its value, this former makes it extremely difficult to use this very powerful method for solving practical problems in this particular case. Moreover, since there does not exist a general constructive technique for matrix factorisation or even for the determination of the partial indices of the matrix, this fact looks like an unavoidable obstacle. Following [2], in this talk, we try to answer a less ambitious question focusing on the determination of the conditions allowing one to construct a family of matrix functions preserving a majority of the properties of the original matrix with non-stable partial indices that is close to the original matrix function.

This work was partially supported by a grant from the Simons Foundation. GM is also acknowledge Royal Society for the Wolfson Research Merit Award.

[1] Gohberg I. & Krein M. 1958 Uspekhi Mat. Nauk.XIII, 3–72 (in Russian).
[2] Mishuris G, Rogosin S. 2018 Regular approximate factorization of a class of matrix-function with an unstable set of partial indices. Proc.R.Soc.A 474:20170279.

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