A matrix equation which appears more suitable than a Sylvester equation in eigenstructure assignment is presented. This equation has easily checkable solvability condition and explicit solution(s). Eigenstructure assignment by state feedback can be solved by using a single matrix equation of this sort, while an additional equation dual to the previous one is needed to meet in the output feedback case. Two strategies, namely, left and right eigenvector assignment and single-side eigenvector assignment, are proposed for solving two coupled matrix equations in the output feedback case. In a matrix algebra setting the discussion on the relationship between the two equations and two well-known Sylvester equations highlights the nature of assigning eigenstructure by output feedback.