We study the orthogonal projections of symplectic balls in ℝ 2 n on complex subspaces. In particular we show that these projections are themselves symplectic balls under a certain complexity assumption. Our main result is a refinement of a recent very interesting result of Abbondandolo and Matveyev extending the linear version of Gromov’s non-squeezing theorem. We use a conceptually simpler approach where the Schur complement of a matrix plays a central role. An application to the partial traces of density matrices is given.