The maximal and minimal ranks and inertias of Y-P were studied by utilizing the maximal and minimal ranks and inertias of linear Hermitian matrix function A-BX-(BX)*,where Y was the leastsquare Hermitian solution of matrix equation AXA*=B,and P was a given Hermitian matrix.The necessary and sufficient conditions of Y>(<,≥,≤)P were obtained.In particular,the maximal and minimal ranks and inertias of Y were given,and equivalent conditions for existence of positive(negative,nonnegative,nonpositive)definite Hermitian leastsquare solution of matrix equation AXA*=B were achieved.