Since the search of the eigenvalues is in general not "simple", equally valid, is the method of reducing with moves of Gauss, that preserve the determinant, (add to multiple rows of other rows, move rows in an even number of times, etc ..) to be traced back to a upper triangular form with zeros below the diagonal. The product of the elements of the diagonal is the determinant, then, if all the elements are greater (resp. lower) than zero then the associated quadratic form is positive definite (resp. negative); if they are greater than or equal (resp. less than or equal) to zero is positive semidefinite (resp. negative semidefinite); undefined if some elements along the diagonal are positive and others negative.
Obviously this is true if the starting matrix is symmetric. What happens if the matrix is not symmetric? Is there a way to study the sign of real part of its eigenvalues not calculate them manually?