A complete analytical solution to the equation and its applications

In this note, a complete, analytical, and restriction-free solution with complete and explicit freedom of the matrix equation is proposed. Here is given and is observable, and is in the Jordan form with arbitrary given eigenvalues. This solution appears to be new because it can be applied directly to obtain significantly better solutions to the following three basic design problems: 1) 2-D system eigenvalue assignment; 2) function observer design; and 3) state feedback eigenstructure design, as shown in this note.