I am reading these notes on viscosity solutions, here is a theorem:
Let us assume $u\in C^2$ is a classical solution of $F(x,u,Du,D^2u)=0$, $x\in \Omega$
then $u$ is a viscosity solution whenever one of the following two is satsified:
1 The PDE does not depend on $D^2u$
2 $F(x,z,p,M)\leq F(x,z,p,N)$ when $M\geq N$
I believe here $M$ and $N$ are Hessian matrices.
My question is what does it mean that a matrix is greater or equal to another?
I remember $M\geq 0$ means it is semi-positive definite, does it mean $M-N$ needs to be semi-positive definite then?