In this paper, a novel expression for the convergence of an iterative learning control algorithm for sampled linear multivariable systems is stated. The convergence analysis shows that, applying this algorithm, the input sequence converges to the system output inverse sequence, specified as a finite-time output trajectory, with zero tracking error on all the sampled points. Also, it gives insight on the learning gain matrix selection to act on the convergence speed or the decoupling of inputs, allowing for an easy tuning using methods from modern control theory. The results are illustrated by some examples, showing a number of options to be investigated.