The Study Of Quaternion Matrix Equation And Quaternion Equality Constrained Least Squares Problem

In this paper,we mainly study the computation of special generalized least squares solution of quaternion linear matrix equation and the quaternion equality constrained least squares problem.For quaternion linear matrix equation AXB(28)C,we transform the problem of computing the least squares bihermitian or skew-bihermitian solution into computing the solution of the corresponding real matrix equation,which use the real representation of quaternion matrix,the structure properties of bihermitian or skew-bihermitian matrix.We propose the expressions of these two kinds of least squares solutions and the corresponding real structure-preserving algorithms.The validity of the algorithms is verified by numerical examples.For quaternion equality constrained least squares problem,by using CS decomposition,QR decomposition,SVD of quaternion matrix and the properties of quaternion MoorePenrose generalized inverse,quaternion orthogonal projection matrix,we prove the equivalence of quaternion equality constrained least squares problem and quaternion Karush-Kuhn-Tucker equation,and obtain equivalent forms of some matrix in the expression of solution of quaternion equality constrained least squares problem.Then using these equivalence,we propose five kinds of method to compute quaternion equality constrained least squares problem: quaternion KKT equation method?unitary decomposition method?Q-SVD method?WLS method and nonconstrained QLS method.We propose real structurepreserving algorithms corresponding to Q-SVD method and WLS method.The validity of the algorithms is verified by numerical examples.