Sturmian theorem are established for weakly coupled elliptic systems generated in a bounded domain by the expressions l 1 u → = − Δ u → + A u → , l 2 w → = − Δ w → + B w → {l_1}\vec u = - \Delta \vec u + A\vec u,{l_2}\vec w = - \Delta \vec w + B\vec w , and Dirichlet boundary conditions. Here Δ \Delta denotes the Laplace operator, and A , B A,B are m × m m \times m matrices. We do not assume that A , B A,B are symmetric, but instead essentially require B B irreducible and b i j ⩽ 0  if  i ≠ j {b_{ij}} \leqslant 0{\text { if }}i \ne j . Estimates on the real eigenvalue of l 2 {l_2} , with a positive eigenvector are then obtained as applications. Our results are motivated by recent theorems for ordinary differential equations established by Ahmad, Lazer and Dannan.