Now we want to maximise g. However, noting that quaternion elements are not independent, the constraint [[File:Equation35.png]] <math>\bar{q}^Tq = 1</math> must also be satisfied. So we use the lagrangian multiplier method and differentiate with respect to [[File:Equation36.png]] <math>\bar{q}</math> we get [[File:Equation37<math>K\bar{q} = \lambda \bar{q}</math>.png]] 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 [[File:Equation37.png]] <math>K\bar{q} = \lambda \bar{q}</math> in [[File:Equation38.png]]<math>g(\bar{q}) = \bar{q}^TKq</math>, we get