Solution of linear matrix equations in a *-congruence class
Roger A. Horn, Vladimir V. Sergeichuk, Naomi Shaked-Monderer · Electronic Journal of Linear Algebra · 2005
The possible *congruence classes of a square solution to the real or complex linearmatrix equation AX = B are determined. The solution is elementary and self contained, and includesseveral known results as special cases, e.g., X is Hermitian or positive semidefinite, and X is realwith positive definite symmetric part.