Finding minima using Lagrange multipliers

Question

I need to minimize the function $ f(A)= a^2+b^2+c^2+d^2$ on the set of matrices with determinant 1.

What I think is that we can take the determinant to be the constraint.

I am new to such type of problems a hint would do.

Thanks & regards

Answer 1

As it was pointed out in the comments, you just need to determine the critical points of the Lagrangian function and select the one that yields the minimum value. This approach works because:

1. the regularity of the objective function and the restriction, together with the fact that the Jacobian matrix of the restrictions has full rank over the feasible set, guarantees that every extrema will be a critical point of the Lagrangian.

2. The fact that the feasible set is closed, together with the interpretation of the objective function as the distance to the origin, guarantees that a minimum exists.