Cell-triangular and cell-diagonal factorizations of cell-triangular and cell-diagonal polynomial matrices
type: Item
instance of: scholarly article
title: Cell-triangular and cell-diagonal factorizations of cell-triangular and cell-diagonal polynomial matrices
zbMATH Open document ID: 0591.15009
DOI: 10.1007/BF01157677
author: V. M. Petrychkovych
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: Let P be a field, \(P[\lambda]\) the ring of polynomials over P, and \(T(\lambda)\in P[\lambda]^{n\times n}\) be an upper-cell triangular, \(\det T(\lambda)\not\equiv 0\). Some sufficient conditions are given for \(T(\lambda)= B(\lambda)C(\lambda)\), where \(B(\lambda)\), \(C(\lambda)\in P[\lambda]^{n\times n}\) are cell-triangular. If \(T(\lambda)\) is cell- diagonal, the author gives also some sufficient conditions for \(T(\lambda)=B(\lambda)C(\lambda)\), where \(B(\lambda),C(\lambda)\in P[\lambda]^{n\times n}\) are cell-diagonal. In addition, the author proves the following result: The matrix T(\(\lambda)\) can be written as a product \(B(\lambda)C(\lambda)\) of \(k\times k\) cell-triangular factors, where the first factor \(B(\lambda)\) is unital of degree s, if and only if the system of matrix equations \[ B_{ii}(\lambda)Y_{ij}(\lambda) + X_{ij}(\lambda)C_{jj}(\lambda) + \sum^{j-1}_{\ell=i+1} X_{i\ell}(\lambda) Y_{\ell j}(\lambda) = T_{ij}(\lambda),\quad 1\leq i<j\leq k \] is solvable.
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A54
zbMATH DE Number: 3948501
zbMATH Keywords: polynomial matrix
zbMATH Keywords: block matrix
zbMATH Keywords: cell-triangular
zbMATH Keywords: cell-diagonal
Wikidata QID: Q114234057
reviewed by: Tong Wenting
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3948501