Full investigation of the matrix equation $AX+XB=C$ and specifically of the equation $AX-XA=C$
E. L. Rabkin · St Petersburg Mathematical Journal · 2014
In the paper, the matrix equations $AX-XA=C$ and $AX+XB=C$ (the Lyapunov equation) are fully investigated and solved if a solution exists. As special cases, exact expressions for the resolvent of the equation $(E-A)X=0$ are obtained for a finite-dimensional operator $A$ and the Fredholm equation of the second kind is studied completely in the finite-dimensional case. The form of a matrix $C$ with nonnegative entries is found such that it is the commutator of a given matrix $A$ with nonnegative entries and some other matrix $X$ with nonnegative entries.