Wiener-Hopf Factorization and Matrix Polynomial Analysis
Explore Wiener-Hopf factorization of matrices and matrix polynomial analysis techniques, including explicit factorization methods and spectral factorization, with a focus on explicit factorization in a vicinity of a given matrix. Learn about the Janashia-Lagvilava method for lower-upper triangular factorization, construction of unitary matrix functions, and the generalization to nxn matrices.
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On explicit Wiener-Hopf factorization of nxn matrices in a vicinity of a given matrix Lasha Ephremidze Razmadze Mathematical Institute of Tbilisi State University, Georgia Kutaisi International University, Georgia October 27 November 1, 2024, Hangzhou, CHINA
Wiener-Hopf factorization: For a given rxr matrix function S(t), defined on the unit circle T in the complex plane, where and can be analytically extended inside T , together with its inverse, can be analytically extended, together with its inverse, outside T . Spectral factorization is a special case where S(t)>0. Partial indices are equal to 0 in this case and is equal to
A brief description of Janashia-Lagvilava method Step 1: The lower-upper triangular factorization of where 2) Construction of unitary matrix function U, such that ?(?)? ? = ? The main computational procedure:
Factorization of matrix polynomial ?(?) with ????(?)=?? 1. Starting point: 2. Intermediate stage: 3. Next step:
A simple example Dilemma: If appearing above is too small should we assume that it arises from round-off errors? We control the growth of the factors during the factorization procedure We can make difference between the 'exact' and regularized factorizations, including the concepts of 'exact' and regularized partial indices.
