Title: Inverting a matrix function around a singularity via local rank factorization
Authors: FRANCHI MassimoPARUOLO PAOLO
Citation: SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS vol. 37 no. 2 p. 774-797
Publisher: SIAM PUBLICATIONS
Publication Year: 2016
JRC N°: JRC94509
ISSN: 0895-4798
URI: http://epubs.siam.org/doi/10.1137/140999839
http://publications.jrc.ec.europa.eu/repository/handle/JRC94509
DOI: 10.1137/140999839
Type: Articles in periodicals and books
Abstract: This paper proposes a recursive procedure, called the extended local rank factorization (elrf), that characterizes the order of the pole and the coefficients of the Laurent series representation of the inverse of a regular analytic matrix function around a given point. The elrf consists in performing a finite sequence of rank factorizations of matrices of nonincreasing dimension, at most equal to the dimension of the original matrix function. Each step of the sequence is associated with a reduced rank condition, while the termination of the elrf corresponds to a full rank condition; this last step reveals the order of the pole. The Laurent coefficients $B_{n}$ are calculated recursively as $B_{n} = C_{n} + \sum_{k=1}^{n} D_{k} B_{n-k}$, where $C_{n}$, $D_{k}$ have simple closed form expressions in terms of the quantities generated by the elrf. It is also shown that the elrf characterizes the structure of Jordan pairs, Jordan chains, and the local Smith form. The procedure is easily cast in an algorithmic form, and a MATLAB implementation script is provided. It is further found that the elrf coincides with the complete reduction process (crp) in Avrachenkov, Haviv, and Howlett [SIAM J. Matrix Anal. Appl., 22 (2001), pp. 1175--1189]. Using this connection, the results on the elrf provide both an explicit recursive formula for $B_{n}$ implied by the crp, and the link between the crp and the structure of the local Smith form.
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