Open AccessSpectral decomposition based solutions to the matrix equation A X − X B = C
Author(s) -
Li ZhaoYan,
Zhou Bin
Publication year - 2018
Publication title -
iet control theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.059
H-Index - 108
eISSN - 1751-8652
pISSN - 1751-8644
DOI - 10.1049/iet-cta.2017.0729
Subject(s) - mathematics , matrix decomposition , matrix (chemical analysis) , matrix difference equation , decomposition , matrix differential equation , riccati equation , mathematical analysis , differential equation , eigenvalues and eigenvectors , materials science , physics , quantum mechanics , composite material , ecology , biology
This study studies the matrix equation A X − X B = 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 A x = 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.