Latest Inversion-Free Iterative Scheme for Solving a Pair of Nonlinear Matrix Equations

Abstract

In this work, the following system of nonlinear matrix equations is considered, X 1 + A ∗ X 1 − 1 A + B ∗ X 2 − 1 B = I and X 2 + C ∗ X 2 − 1 C + D ∗ X 1 − 1 D = I , where A , B , C , and D are arbitrary n × n matrices and I is the identity matrix of order n . Some conditions for the existence of a positive-definite solution as well as the convergence analysis of the newly developed algorithm for finding the maximal positive-definite solution and its convergence rate are discussed. Four examples are also provided herein to support our results.

Inverse Eigenvalue Problem and Least‐Squares Problem for Skew‐Hermitian {P,K + 1}‐Reflexive Matrices

“…For the matrix equation AX � B, the Hermitian reflexive, antireflexive, generalized reflexive, and antireflexive solutions have been discussed in [15,16]. And different solution sets for other equations have also been discussed in [17][18][19][20]. Besides, the optimization problems of different matrices have also been studied in [21][22][23][24].…”