Normalisation of Wiener--Hopf Factorisation for Matrix Functions
Abstract
{For a matrix function A(t), we study the normalisation of its Wiener--Hopf factorisation A(t)=A-(t)D(t)A+(t) which guarantees the uniqueness of the factors A±(t). Previously, this problem was investigated for second-order matrix functions, the stable factorisation, and the factorisation of matrix functions with distinct partial indices. The main tools for solving the problem of normalisation are the theorem of I.C. Gohberg and M.G. Krein on a general form of the factors A±, a Birkhoff factorisation of matrix functions, and a block PLU factorisation of the numerical matrix A-(∞). To solve the normalisation problem, a concept of P-normalised factorisation is introduced. The Birkhoff factorisation of matrix functions plays a key role in defining this concept. The permutation matrix P is determined by PLU factorisation of the numerical matrix A-(∞). We proved that any invertible matrix function A(t) admits P-normalisation. This normalisation guarantees the uniqueness of the Wiener--Hopf factorisation. The P-normalised factorisation of A(t) allows us to find the Birkhoff factorisation of A(t). Without normalisation, it is impossible to obtain explicit estimates of the absolute errors in the approximate computation of the factorisation factors.}
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BibTeX
@misc{adukova2026normalisation,
title = {{Normalisation of Wiener--Hopf Factorisation for Matrix Functions}},
author = {Nataliia Adukova and Victor Adukov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07957},
eprint = {2610.07957},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07957},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Normalisation of Wiener--Hopf Factorisation for Matrix Functions
AU - Nataliia Adukova
AU - Victor Adukov
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07957
UR - https://arxiv.org/abs/2610.07957
AB - {For a matrix function A(t), we study the normalisation of its Wiener--Hopf factorisation A(t)=A_-(t)D(t)A_+(t) which guarantees the uniqueness of the factors A_±(t). Previously, this problem was investigated for second-order matrix functions, the stable factorisation, and the factorisation of matrix functions with distinct partial indices. The main tools for solving the problem of normalisation are the theorem of I.C. Gohberg and M.G. Krein on a general form of the factors A_±, a Birkhoff factorisation of matrix functions, and a block PLU factorisation of the numerical matrix A_-(∞). To solve the normalisation problem, a concept of P-normalised factorisation is introduced. The Birkhoff factorisation of matrix functions plays a key role in defining this concept. The permutation matrix P is determined by PLU factorisation of the numerical matrix A_-(∞). We proved that any invertible matrix function A(t) admits P-normalisation. This normalisation guarantees the uniqueness of the Wiener--Hopf factorisation. The P-normalised factorisation of A(t) allows us to find the Birkhoff factorisation of A(t). Without normalisation, it is impossible to obtain explicit estimates of the absolute errors in the approximate computation of the factorisation factors.}
ER -