If we have a second order quasilinear PDE of the form
$A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$
where $A,B,C$ are functions of $x,y,u$,
then the equation is called elliptic if $det=\begin{vmatrix}A &C \\C & B\end{vmatrix}>0$, parabolic if $det=0$ and hyperbolic if $det<0$.
Now what happens if we have a system of two coupled PDEs of the form
$A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+D\frac{\partial^2 v}{\partial x^2}+E\frac{\partial^2 v}{\partial y^2}+2F\frac{\partial^2 v}{\partial x\partial y}+ lower\,\, order \,\, terms=0 \\ G\frac{\partial^2 u}{\partial x^2}+H\frac{\partial^2 u}{\partial y^2}+2K\frac{\partial^2 u}{\partial x\partial y}+L\frac{\partial^2 v}{\partial x^2}+M\frac{\partial^2 v}{\partial y^2}+2N\frac{\partial^2 v}{\partial x\partial y}+ lower\,\, order \,\, terms=0$
with $A,B,C,...$ being functions of $x,y,u,v$.
Does it make sense to construct the determinant
$det=\begin{vmatrix}A &C & D&F\\C & B&F&E\\G &K & L&N\\K &H & N&M\end{vmatrix}$
and investigate its sign, or something like that?
--------- Update ----------
In the chapter 10 of the book [Seiler, Werner M., Involution. The formal theory of differential equations and its applications in computer algebra] the author states that since ellipticity is a property defined at points and depends only on the principal symbol it then suffices to study it for linear equations. I understand that the same procedure holds for quasilinear ones, the only difference is that the coefficients are allowed to be functions of the dependent variables. Then regarding the specific example above if one implement the theory for linear equations the principal symbol should be
$\tau[\xi,\eta]=\begin{bmatrix}B \eta^2 + 2 C \eta\xi + A \xi^2& E\eta^2 + 2 F \eta\xi + D \xi^2\\ H\eta^2 + 2 K \eta\xi + G \xi^2 & M\eta^2 + 2 N \eta\xi + L \xi^2\end{bmatrix}$
Ellipticity requires $det(\tau)\neq 0$ that is
$(B M-E H) \eta^4 + 2 (C M + B N-F H - E K ) \eta^3 \xi + (B L + A M + 4 C N-E G - D H - 4 F K ) \eta^2 \xi^2 + 2 ( C L + A N-F G - D K ) \eta \xi^3 + (A L-D G ) \xi^4\neq 0$
well, when this condition is to be satisfied? Is it true that it is hyperbolic in direction $x$ or $y$ direction, if
$(B M-E H) \eta^4 + 2 (C M + B N-F H - E K ) \eta^3 \xi + (B L + A M + 4 C N-E G - D H - 4 F K ) \eta^2 \xi^2 + 2 ( C L + A N-F G - D K ) \eta \xi^3 + (A L-D G ) \xi^4 = 0$
possesses only simple real roots?
