Quadratic equations over matrix skew series (Q463765)

The paper deals with the solutions to a general matrix quadratic equation (GQME) \[ \sum_{u=1}^U A_u X B_u X C_u+\sum_{v=1}^V D_v X E_v + F=0, \] where all \(A_u,B_u,C_u, D_v,E_v,F,X\) are square \(n\times n\) matrices. The general component-wise solution of the GQME means solving \(n^2\) scalar equations whose degree vary from 2 to \(2^{n^2}\). Thus, the author presents conditions when the GQME can be solved in a feasible manner. The key is a decomposition of a given matrix into a set of (upper-, lower-) diagonal or anti-diagonal matrices (skew series, dual-skew series). Special care is devoted to diagonal, anti-diagonal, triangular matrices entering the GQME and their block versions; these cases can be replaced by a system of scalar equations.