Cell-triangular and cell-diagonal factorizations of cell-triangular and cell-diagonal polynomial matrices (Q1074687)

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.