This paper is concerned with the positive definite solutions to the matrix equation X + A H X −1 A = I where X is the unknown and A is a given complex matrix. By introducing and studying a matrix operator on complex matrices, it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation W + B T W −1 B = I which has been extensively studied in the literature, where B is a real matrix and is uniquely determined by A. It is also shown that if the considered nonlinear matrix equation has a positive definite solution, then it has the maximal and minimal solutions. Bounds of the positive definite solutions are also established in terms of matrix A. Finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed.
In this study, a novel functionalized graphene oxide (F-GO) modified polyamide 6 (PA6) sizing agent is developed to improve the interfacial and mechanical properties of carbon fiber (CF) reinforced PA6 composites. To enhance compatibility between GO and sizing agent, PA6 molecular chains are covalently grafted onto the GO sheets in advance. The results show that the F-GO modified sizing layer improves interfacial compatibility and wettability of the CF surface. Moreover, F-GO nanoparticles retard crack propagation along the interface based on the reinforcement effect. Accordingly, the interfacial shear strength increases by 54.5% from 45.1 MPa for desized CF/PA6 composites to 69.7 MPa for F-GO modified sizing agent treated CF/PA6 composites. Meanwhile, the average interlaminar shear strength, tensile strength, and flexural strength increase by 45.2%, 15.6%, and 19.3% from 45.7, 1307.4, and 792.2 MPa for desized CF/PA6 composites to 66.3, 1511.2, and 945.2 MPa for those reinforced with the CFs treated by the F-GO modified sizing agent due to the effective stress transfer across the interface. The improvement in the mechanical properties is verified based on the cross-section morphology and fracture surfaces.
Periodic Lyapunov matrix equations play important roles in stability analysis and stabilisation of discrete-time periodic systems. In this paper, an iterative algorithm for solving periodic Lyapunov matrix equations is established. It is shown that the proposed iteration converges to the unique solution of the considered matrix equations at finite steps with arbitrary initial condition despite the stability of the associated discrete-time periodic linear systems. Both the reverse-time and forward-time discrete-time periodic Lyapunov matrix equations are considered. Numerical examples are worked out to illustrate the effectiveness of the proposed algorithm.
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