In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if the domain is defined by a $C^\infty$ defining function $\phi>0$ , then 'Levi pseudoconvex ' is equivalent to the following matrix (which is called 'Monge-Ampere matrix')
\begin{equation*}
\left(
$$ \begin{array}{cc}
begin{pmatrix} -\phi, \phi & -\partial_\bar{k}\phi ; \\
-\partial_j\phi, \partial_j\phi & -\partial_{j\bar{k}}^2\phi \
\end{array}
\right)
\end{equation*}
end{pmatrix} $$
have precise one negative eigenvalue and n positive eigenvalues .
So my question is how to prove this ? Anybody knows? Thanks very much!
