Zhuo-Heng He, Oscar Mauricio Agudelo, Qing-Wen Wang, Bart De Moor, Two-sided coupled generalized Sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices, Linear Algebra and its Applications, Volume 496, 1 May 2016, Pages 549–593, doi:10.1016/j.laa.2016.02.013.

**Abstract: **In this paper, we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices: A_{i}∈C^{pi×ti}Ai∈Cpi×ti, B_{i}∈C^{si×qi}Bi∈Csi×qi, C_{i}∈C^{pi×ti+1}Ci∈Cpi×ti+1, D_{i}∈C^{si+1×qi}Di∈Csi+1×qi, and E_{i}∈C^{pi×qi}Ei∈Cpi×qi (i=1,2,3i=1,2,3). We show that from this simultaneous decomposition we can derive some necessary and sufficient conditions for the existence of a solution to the system of two-sided coupled generalized Sylvester matrix equations with four unknowns A_{i}X_{i}B_{i}+C_{i}X_{i+1}D_{i}=E_{i}AiXiBi+CiXi+1Di=Ei (i=1,2,3i=1,2,3). Apart from proving an expression for the general solutions to this system, we derive the range of ranks of these solutions using the ranks of the given matrices A_{i}Ai, B_{i}Bi, C_{i}Ci, D_{i}Di, and E_{i}Ei. We provide some numerical examples to illustrate our results. Moreover, we present a similar approach to consider the simultaneous decomposition for 5*k* matrices and the system of *k * two-sided coupled generalized Sylvester matrix equations with k+1k+1 unknowns A_{i}X_{i}B_{i}+C_{i}X_{i+1}D_{i}=E_{i}AiXiBi+CiXi+1Di=Ei (i=1,…,ki=1,…,k, k≥4k≥4). The main results are also valid over the real number field and the real quaternion algebra.

*Source:* http://www.sciencedirect.com/science/article/pii/S0024379516001142

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