Assume that the second order constant coefficient linear differential equation
$\frac{\partial^2 u}{\partial x_1^2} + 2b \frac{\partial^2 u}{\partial x_1 \partial x_2} + c \frac{\partial^2 u}{\partial x_2^2} = 0$, the corresponding matrix $M$ is specified as $\begin{pmatrix} a & b \\ b & c \end{pmatrix}$. By variable
substitution $\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = P \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$, where $P = \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix}$, the equation can be transformed into
$\frac{\partial^2 u}{\partial y_1^2} + 2b' \frac{\partial^2 u}{\partial y_1 \partial y_2} + c' \frac{\partial^2 u}{\partial y_2^2} = 0$, with the corresponding matrix.
$M' = \begin{pmatrix} a' & b' \\ b' & c' \end{pmatrix}$.
1. For general cases, prove that $M' = PMP^T$. Supposed $a \frac{\partial^2 u}{\partial x_1^2} + 2b \frac{\partial^2 u}{\partial x_1 \partial x_2} + c \frac{\partial^2 u}{\partial x_2^2} = 0$ is hyperbolic,
what type of equation is obtained after coordinate transformation?
