On The Solutions Of Three Kinds Of Matrix Equations

Linear matrix equations play an important role in structural design,model modification,control theory and so on.In this paper,the solutions of three kinds of matrix equations are considered:the Re-nonnegative definite(Re-nnd)and Re-positive definite(Re-pd)solutions to the matrix equations A1XA1*=C1,A2XA2*=C2,where A1?Cm×n,A2?Cp×n,C1?Cm×m?C2?Cp×p;the generalized inverse eigenvalue problem of Hamiltonian matrices and its approximation,that is,given ?=diag(?1,…,?p)?Cn×p,X=[x1,…,xp]?C2n×p,A,B?R2n×2n,find(A,B)?S1 such that ?A-A?2+?B-B?2=(?)(?A-A?2+?B-B?2),where S1={(A,B)?HR2n×2n×HR2n×2n|AX?=BX};the model updating problem of the piezoelectric smart structure,that is,given ?=diag(?1,…,?p)?Rp×p,Z=[Z1T,Z2T]T?Rn×p,(?)find(M,K)?S2 such that ?M-MA?2+?K-KA?2=(?)(?M-MA?2+?M-KA?2),where#12By applying the Moore-Penrose inverses and the spectral decompositions of matrices,a new method to solve the Re-nnd and Re-pd solutions of the matrix equations A1XA1*=C1,A2XA2*=C2 is proposed,and the solvability conditions and the explicit representation of the general solution of the matrix equations are obtained.Then,the numerical algorithm and numerical examples are given,and the correctness of the results is verified by numerical examples.By using QR-decomposition of modal matrix,a direct method to solve the generalized inverse eigenvalue problem of Hamiltonian matrices is proposed,and the general expression of the inverse problem is proposed.Then,the optimal approximation solution of the known matrix pair is given by applying the optimal approximation theorem and matrix derivative.Furthermore,the numerical algorithm and numerical example are given,and the feasibility of the method is verified by numerical example.By using the generalized singular value decomposition of a matrix pair,a direct method to solve the model updating problem of the piezoelectric smart structure is proposed,and the solvability condition and the explicit representation of the general solution of the problem are obtained.Then,the optimal approximation solution of the known matrix pair is given by applying the matrix derivatives.Furthermore,the numerical algorithm and numerical example are given,and the effectiveness of the method is verified by numerical example.