3 fixed math display

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!

2 added 3 characters in body

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} -\phi \phi, & -\partial_\bar{k}\phi ; \ -\partial_j\phi \partial_j\phi, & -\partial_{j\bar{k}}^2\phi \ \end{array} \right) \end{equation*} have precise one negative eigenvalue and n positive eigenvalues .

So my question is how to prove this ? Anybody knows? Thanks very much!

1

# how to prove the relationship between pseudoconvexity and the monge-ampere matrix?

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} -\phi & -\partial_\bar{k}\phi \ -\partial_j\phi & -\partial_{j\bar{k}}^2\phi \ \end{array} \right) \end{equation*} have precise one negative eigenvalue and n positive eigenvalues .

So my question is how to prove this ? Anybody knows? Thanks very much!