This system is not going to be square in general, as for each measurement being made there will generally be multiple measurement outcome projectors E i . For example, in a 2-D Hilbert space with 3 measurements σ σ x , σ σ y , σ σ z , each measurement has 2 outcomes, each of which has a projector E i, for 6 projectors, whereas the real dimension of the space of density matrices is (2⋅2)/2=4, leaving A to be 6 x 4. To solve the system, multiply on the left by A T :