The elliptic ball criterion is proved for the discrete-time non-linear feedback control equation p(6) x+ BM(t)q(B)x = 0, in which θxlpar;t) = x( t + 1). It is a geometrical condition for stability and instability which reduces to the well-known circle criterion in. the special case when the matrix M(t) is a scalar. For a certain class of equations the elliptic ball criterion is shown to be both necessary and sufficient for the existence of a special kind of quadratic Lyapunov function. The analogy with differential equations is found to fail in one respect