The Hankel Matrix Solution to a System of Quaternion Matrix Equations

摘要

The solution of matrix equations and the optimal approximation problem play an important role in linear optimal control, parameter identification, structural vibration, aviation and other fields. Hankel matrix is kind of matrix with special structure and wide application. In this paper, the problem of Hankel constraint solution to the system [AXB CXD]=[E F] over quaternion field is discussed. By using the representation of vectors of a Hankel matrix and Kronecker product of matrices, a constrained problem will be transformed into an unconstrained equation. Then the necessary and sufficient conditions for the equations with Hankel solution as well as the expression of general solution are obtained. Meanwhile, when the solution set is nonempty, by using invariance of Frobenius norm of orthogonal matrix product, the optimal approximation solution with minimal Frobenius norm for a given Hankel matrix is derived. Finally, two numerical examples is provided to verify the algorism.