It certainly doesn't. What you want to do is to maximise the Σ-norm of BX subject to the Σ-norm of X being at most 1. If

then

. So you want to maximise

subject to

. That looks quite unpleasant. But suppose we cheat a bit and assume that w, x, y and z are all positive. Then the problem becomes: maximise 4w+4x+6y+6z subject to w+x+y+z=1, and the maximum is easily seen to be 6. I would guess that this is the correct answer to the problem, but I don't offhand see how to justify that (without making the cheating assumption).