Nonnegative-definite Solutions To The Morphism Equations αxα～*=β And αxβ=γ

In this paper, we give the sufficient and necessary condition for the existence of the nonnegative-definite solution to the morphism equation αx α = β which is that the equation αy = β is consistent.In this case, a representation of the general solution is also gived :In the next two corollaries ,we study the structure of the general solution carefully; in the third corollary, we give the sufficient and necessary condition for the existence of the positive-definite solution.Applying these results to the matrix equation AXA* = B over /7-division ring, we get the sufficient and necessary condition for the existence of the nonnegative-definite solution to the matrix equation AXA* = B which is that the equation AY = B is consistent. Similarly, a representation for the general solution is gived :By the properties of matrix and the structure of the general solution, we obtain the formula for the rank of X and the representations for the general nonnegative-definite solutions of minimal and maximal rank respectively.Adding a property to the category, we attain the sufficient arid necessary condition for the existence of the nonnegative-definite solution to the morphism equation αxβ = γ for some special morphisms α,β,γ, followed by a representation of the general solution. As their application, we give some relevant results of the matrix equation AXB = C overp-division ring.