From ad28c073ec655a16c428c8df90859e624ce2addd Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 16:14:27 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/report.tex b/report.tex
index a30296f..e92c068 100644
--- a/report.tex
+++ b/report.tex
@@ -246,6 +246,12 @@
       \pdvt{u^I} + \tensor{A}{^i^I_J} \pdv{u^J}{x^i} = S^I,
     \end{equation}
     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 $P(\xi) := \tensor{A}{^i^I_J} \xi_i$.
+    The system is said to be:
+    \begin{itemize}
+      \item \emph{weekly hyperbolic} if for any $\xi_i$, $P(\xi)$ has only real eigenvalues.
+      \item \emph{strongly hyperbolic} if for any $\xi_i$, $P(\xi)$ 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 $H_{IK} \tensor{A}{^i^K_J}$ is symmetric for each $i$.
+    \end{itemize}
   \end{frame}
 
   \begin{frame}{Existing first order formulations}