[[File:Equation33.png|frame|center]]

Wherein,

Now we want to maximise g. However, noting that quaternion elements are not independent, the constraint <math>\bar{q}^Tq = 1</math> must also be satisfied. So we use the lagrangian multiplier method and differentiate with respect to <math>\bar{q}</math> we get <math>K\bar{q} = \lambda \bar{q}</math>. So now, we have reduced the problem to an eigenvalue problem for the matrix K.

Since, K is a 4x4 matrix, it can have at most 4 different eigenvalues. Substituting <math>K\bar{q} = \lambda \bar{q}</math> in <math>g(\bar{q}) = \bar{q}^TKq</math>, we get