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Cell-triangular and cell-diagonal factorizations of cell-triangular and cell-diagonal polynomial matrices. (English. Russian original) Zbl 0591.15009
Math. Notes 37, 431-435 (1985); translation from Mat. Zametki 37, No. 6, 789-796 (1985).
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.
Reviewer: Wenting Tong

##### MSC:
 15A23 Factorization of matrices 15A54 Matrices over function rings in one or more variables
##### Keywords:
polynomial matrix; block matrix; cell-triangular; cell-diagonal
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##### References:
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