In this paper, a MIMO control law for minimizing a new H₂ index is proposed. The controller is given in closed mathematical form, is strongly stabilizing and has zero dc gain, resulting in a bandpass structure useful in some applications (e.g. flexible systems employing accelerometers to sense vibrations). As the above control strategy is deduced in scaled form, in this paper the selection of the scaling input and output matrices is also analyzed, by minimizing a new index proposed to increase the effectiveness of the control action in minimizing an H₂ index. Although the problem turns out to be a nonlinear programming problem, a two-stages algorithm is provided for computing the optimal solution, based on explicit formulas for the computation of the H₂ index and on the solution of a generalized eigenvalue problem. The algorithm allows the selection of the direction of the scaling matrices, while their norm can be increased at will in order to increase the control authority. Numerical simulations show the effectiveness of the proposed approach.