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Yingjie Wang
2026-04-09 17:01:48 -04:00
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where both $A$ and $S$ can depend on $u^I$ but not their derivatives. For any unit covector $\xi_i$, the \emph{Principal symbol} matrix is defined as $\tensor{P}{^I_J}(\xi) := \tensor{A}{^i^I_J} \xi_i$. where both $A$ and $S$ can depend on $u^I$ but not their derivatives. For any unit covector $\xi_i$, the \emph{Principal symbol} matrix is defined as $\tensor{P}{^I_J}(\xi) := \tensor{A}{^i^I_J} \xi_i$.
The system is said to be: The system is said to be:
\begin{itemize} \begin{itemize}
\item \emph{weekly hyperbolic} if for any $\xi_i$, $\tensor{P}{^I_J}(\xi)$ has only real eigenvalues. \item \emph{weakly hyperbolic} if for any $\xi_i$, $\tensor{P}{^I_J}(\xi)$ has only real eigenvalues.
\item \emph{strongly hyperbolic} if for any $\xi_i$, $\tensor{P}{^I_J}(\xi)$ has only real eigenvalues and a complete set of eigenvectors, i.e., it is diagonalizable. \item \emph{strongly hyperbolic} if for any $\xi_i$, $\tensor{P}{^I_J}(\xi)$ has only real eigenvalues and a complete set of eigenvectors, i.e., it is diagonalizable.
\item \emph{symmetrically hyperbolic} if there exists a positive definite symmetrizer matrix $H_{IJ}$ such that $H_{IK} \tensor{P}{^K_J}(\xi)$ is always symmetric. \item \emph{symmetrically hyperbolic} if there exists a positive definite symmetrizer matrix $H_{IJ}$ such that $H_{IK} \tensor{P}{^K_J}(\xi)$ is always symmetric.
\end{itemize} \end{itemize}