A quaternion antisymmetric and persymmetric matrix inverse problem from Hopfield neural networks

This paper considers the antisymmetric and per-symmetric solution to a matrix inverse problem AX = B for A and the optimal approximation over the quaternion field H. We first give the specified structure of the antisymmetric and persymmetric quaternion matrix. Then we derive the necessary and sufficient conditions for the existence of and the general expression for the antisymmetric and persymmetric solution of the matrix equation mentioned above. Moreover, we obtain the expression of the solution to optimal approximation problem and corresponding numerical algorithm is also presented. The work is motivated and illustrated with a problem of Hopfield neural networks.