Given a fundamental matrix ϕ ( x ) of an n -th order system of linear homogeneous differential equations Y ′ = A ( x ) Y , a necessary and sufficient condition for the existence of a k -dimensional ( k ≤ n ) periodic sub-space (of period T ) of the solution space of the above system is obtained in terms of the rank of the scalar matrix ϕ ( t ) − ϕ ( 0 ) .