The eigenvalues range of a class of matrices and some applications in Cauchy-Schwarz inequality and iterative methods

This paper discusses the range of the eigenvalues of a class of matrices. By using the eigenvalues range of a class of matrices, an extension of the inner product type Cauchy-Schwarz inequality is obtained, the convergence proof of the least squares based iterative algorithm for solving the coupled Sylvester matrix equations is given and the best convergence factor is determined. Moreover, by using the eigenvalues range of this class of matrices, an iterative algorithm for solving linear matrix equation is established. Three numerical examples are offered to illustrate the effectiveness of the results suggested in this paper.