The Re—Positive Semidefinite Solution to the Quaternion Matrix Equation AX=B

An n×n real quaternion matrix A is termed Re-psd(or Re-pd)if the real part of xAx is nonegative(or positive)for every nonzero quaternion n-vector x.The necessary and suf-ficient condition for the existence of and the expression for Re-psd(Re-pd)solution of the lin-ear mat ix equation AX=B over the quaternion field are derived.In addition,a criterion for apartitioned n×n quaternion matrix is Re-psd(or Re-pd)is presented.