Spectral decomposition based solutions to the matrix equation AX - XB = C

This study studies the matrix equation AX - XB = C, which has many important applications in control theory, by using spectral decompositions of A and B. By establishing solvability conditions and solutions to the standard vector equation Ax = c, and the spectral decompositions of the associated nivellateur of the matrix equation, necessary and sufficient conditions for the solvability of the matrix equation are provided in terms of the coefficients of the spectral decompositions of A and B. Moreover, explicit solutions are provided, which are also based on the coefficients of the spectral decompositions of A and B. The obtained results include the existing ones as special cases, and, moreover, correct some errors in the existing methods. The effectiveness of the proposed approach is demonstrated by some illustrative examples.