Do positive semidefinite matrices have to be symmetric?

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!

An example of a non-symmetric positive definite matrix is $$M=\pmatrix{2&0\\2&2}.$$ Indeed, $$\pmatrix{x\\y}^T\pmatrix{2&0\\2&2}\pmatrix{x\\y} = (x+y)^2 + x^2 + y^2$$ which is strictly greater than $0$ whenever the vector is non-zero.