Generally you want to inform Julia that a matrix is symmetric by wrapping it in Symmetric
, e.g.
A = Symmetric(M + M')
although it appears that, as you say, the symmetry is being recognized and exploited in this case.
As to the speed difference, if you are running using an Intel CPU, it could be due to the fact that your version of NumPy is compiled with MKL, while Julia, by default, uses OpenBlas. Try
using MKL
prior to your other code and see if it makes a difference.