Is Symmetry Required for Determining the Hamiltonian?

[aaaa202]

My book writes a 5-step recipe for detemining the hamiltonian, which I have attached. However I see a problem with arriving at the last result. Doesn't it only follow if the matrix M is a symmetric matrix - i.e. the transpose of it is equal to itself.

 

[praharmitra]

I am assuming the tilde above an object implies taking a transpose. If that is the case, then M is indeed a symmetric matrix. One can see this by looking at the Lagrangian. Since the Lagrangian is a number, we can take a transpose of L and we'll get back the same number, i.e. L^T=L, this will rewrite the Lagrangian in terms of M^T. But since these are equal, we must have M^T=M