More A_{1}E+EA_{2}=-D and X=A_{1}X+XA_{2}+D,X(0)=C
Richard T. Lacoss, A. Shakal · IEEE Transactions on Automatic Control · 1976
The solution of the real matrix equationA_{1}E + EA_{2} = - Dis given in terms of the eigenvalues and eigenvectors ofA_{1} \in R^{{n}_{1} \times n_{1}}, A_{2} \in R^{n_{2}} \times n_{2}}, and the elements ofD. The solution is valid if the Aiare symmetric or if they do not have repeated eigenvalues. The solution can be extended to handle nonsymmetric matrices with repeated eigenvalues. The solution of\dot{X} = A_{1}X+XA_{2} + D, X(0) = Cis also given in closed form for the case of nonrepeated eigenvalues or symmetric Ai.