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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?

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Yes, exactly. $ $ – Berci Apr 24 '13 at 9:55
It's most likely related to semi-positive definiteness, as you mention. But we'll need to know more about the context to give a definitive answer. – Raskolnikov Apr 24 '13 at 9:55
@Raskolnikov let me add the full statement of the theorem. – Lost1 Apr 24 '13 at 9:59
Minor language quibble: I think it is more common to say positive semidefinite than semi-positive definite. – Harald Hanche-Olsen Apr 24 '13 at 11:08
up vote 2 down vote accepted

To the (admittedly quite limited) extent that I know the literature on viscosity solutions, your interpretation is the correct one.

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From my knowledge, when matrix A is greater than matrix B, it means that A-B is positive definite.

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