The Inverse Quadratic Eigenvalue Problem For Two Special Classes Of Matrices

The matrix eigenvalue inverse problem refers to determination of matrix elements according to the information of eigenvalues or / and eigenvectors under certain constraints.The quadratic eigenvalue inverse problem refers to the problem which is opposite to the quadratic eigenvalue frequently encountered in engineering technology,especially in the field of structural dynamic model modification.In this paper,the quadratic eigenvalue inverse problems of two special matrices are studied.In other words,the special forms of eigenvalues,eigenvectors and coefficient matrices are given for matrix equations.In this way,the solution set and the best approximation solution of matrix equations satisfying these conditions are obtained.In this paper,the quadratic eigenvalue inverse problems of two kinds of special matrices are studied systematically.The first type is the quadratic eigenvalue inverse problems of symmetric orthogonal symmetric matrices and their best approximations.The second type is the quadratic eigenvalue inverse problems of symmetric subantisymmetric matrices and their best approximations.In view of the first type,the general problem of quadratic eigenvalue inverse problem of symmetric orthogonal symmetric matrix is given.According to the property of symmetric orthogonal symmetric matrix,the coefficient matrix is divided into blocks,and the equation is transformed into equivalent matrix equations.By singular value decomposition and QR decomposition,the set of orthogonal symmetric solutions for general problems is obtained.According to the unitary invariance of Frobenius norm and optimization theory,the optimal approximation equation is simplified.Finally,the symmetric orthogonal symmetric solutions and the optimal approximations of quadratic eigenvalue inverse problems under two decomposition conditions are obtained.Aiming at the second type,the general problem of quadratic eigenvalue inverse problem of symmetric subantisymmetric matrices is firstly given.According to the property of symmetric subantisymmetric matrices,the coefficient matrices are partitioned,and then the equations are transformed into equivalent matrix equations.By the decomposition of QR,the set of symmetric subantisymmetric solutions for general problems is obtained.According to the unitary invariance of Frobenius norm and optimization theory,the optimal approximation equation is simplified.Finally,the symmetric subantisymmetric solution of quadratic eigenvalue inverse problem and its optimal approximation are obtained.