The Inverse Problem Of Quadratic Eigenvalues Of Centrosymmetric Matrices Under The Constraint Of Main Subarray

The matrix eigenvalue inverse problem,also known as the constrained matrix equation problem,refers to the solution of a given matrix equation in a set of matrices under certain constraints.This paper mainly studies the quadratic eigenvalue inverse problem and its optimal approximation under the constraints of the main sub-matrix,that is,taking the matrix equation(?) as the main research content,to constrain the main sub-matrix of its coefficient matrices,M,Cand,K and obtain the solutions to matrix equations.This paper systematically studies two types of inverse quadratic eigenvalue problems under the main sub-matrix constraint.The first type is the inverse quadratic eigenvalue problem of a symmetric matrix under the main sub-matrix constraint and its best approximation solution in the solution set.The two types are the inverse quadratic eigenvalue problems of centrosymmetric matrices under the constraint of the principal sub-matrix and their best approximation solutions in the solution set.For the first type,the coefficient matrix is divided into blocks based on the properties of the symmetric matrix,and then the special properties of the block matrix are used to find the equivalent equations of the inverse quadratic eigenvalue problems on the given set respectively,and then obtain the The necessary and sufficient conditions for the problem to be solved and the general form of the general solution.For the optimal approximation problem,the optimal approximation solution can be obtained through the special properties of the block matrix,the unitary invariance of the F norm of the matrix and the optimization theory to simplify the optimal approximation equation,and the solution is proved to exist and be unique.For the second type,the block expression of the matrix is obtained according to the properties of the central symmetric matrix,and the sufficient and necessary conditions for the solution of the general matrix equation are used to obtain the solvable conditions and general solution expressions of the problem.In the method,the main application is The properties of special matrices,matrix block and singular value decomposition are discussed.For the optimal approximation problem,the optimal approximation solution can be obtained by giving the lemma,the unitary invariance of the matrix F norm and the optimization theory to simplify the optimal approximation equation,and it is proved that the solution exists and is unique.The matrix quadratic eigenvalue inverse problem is a very important problem.For example,in the field of engineering technology,especially in the field of structural dynamic model correction technology,the problem that is opposite to the quadratic eigenvalue problem is often encountered.Therefore,this subject mainly studies the quadratic eigenvalue inverse problem under constraints and its optimal approximation.At present,this problem has been applied to the fields of structural design,inverse problem of damped vibration systems and so on.