The Solutions of a Class of Sylvester-like Linear Matrix Equations and the Estimation of the Associated Measurements of Their Solutions

This paper studies solutions and relevant measures of a class of Sylvester-like linear matrix equations commonly encountered in control theory. Firstly, inequalities related to the singular values of solutions of a class of Sylvester-like linear matrix equations are obtained. These results improve upon existing relevant studies. Next, starting from the definition of singular values for any matrix, a lower bound for the product of solutions and their complex conjugate transpose matrices is directly obtained. Additionally, when a Hermite matrix is a solution to the matrix equation, a convergent matrix series is obtained, as the positive definite solution under certain conditions. Finally, we design two algorithms for solving the class of matrix equations, where each recursive iteration results in obtaining the upper and lower solution bounds. Numerical experiments demonstrate that our results outperform some existing studies.