TITLE: Do positive semidefinite matrices have to be symmetric?
QUESTION [27 upvotes]: Do positive semidefinite matrices have to be symmetric? Can you have a non-symmetric matrix that is positive definite? I can't seem to figure out why you wouldn't be able to have such a matrix, but all my notes specify positive definite matrices as "symmetric $n \times n$ matrices."
Can anyone help me with an example of a non-symmetric positive definite matrix, or some insight into a proof for why it would need to be symmetric should that be the case? Thanks!

REPLY [8 votes]: Let me just add that there exist a branch of optimization and variational analysis in which the notion of a nonnegative definite matrix which is not symmetric is fundamental. Consider a smooth convex function; by convexity its Hessian is nonnegative positive definite while by Schwarz Theorem, it is also symmetric. However, there exist "non-gradient operators" i.e. which do not integrate into a function but still possess a notion of convexity. This is the notion of a monotone operator, which in the smooth case, its Jacobian is a nonnegative definite matrix which may not be symmetric.