Let be the space of symmetric matrices and is the orthogonal matrices. Suppose satisfies

,

Let satisfy

, , and (1)

The fully nonlinear equation is called elliptic in if

for any (2).

From (1), we know there exists such that , where are the eigenvalues of . And must be symmetric.

is elliptic is equivalent to ,

Firstly, let us assume is elliptic, i.e. , .

Then in particular choose ,

Then

Secondly, assume , .

, there exists such that , here we denote .

Then

Differentiating (1), we get

That is

Since , then ,

For , when , by definition and **use the fact that** (possible to remove?)

here is symmertric and has entries which are 0 except at -th and -th positions are 1.

has eigenvalue with and , .

So

- If , then (4) becomes

because is symmetric and .

- If , (4) becomes

Combing the above results, we get .

Applying to (3), it becomes

This means .

is a positive definite matrix because , .