MoreA_{1}E+EA_{2}=-DandẊ=A_{1}X+XA_{2}+D,X(0)=C

The solution of the real matrix equation
A_{1}E + EA_{2} = - D
is given in terms of the eigenvalues and eigenvectors of
A_{1} \in R^{{n}_{1} \times n_{1}}, A_{2} \in R^{n_{2}} \times n_{2}}
, and the elements of
D
. The solution is valid if the A
i
are 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) = C
is also given in closed form for the case of nonrepeated eigenvalues or symmetric A
i
.