update: auto commit

report.tex

@@ -123,7 +123,7 @@
 
     General $N$ black hole puncture initial data:
     \begin{equation*}
-      \psi = \sum_{i=1}^N \frac{M_i}{2|\vb*{r}-\vb*{r}_i|} + (\text{regular part}),
+      \psi = \sum_{i=1}^N \frac{M_i}{2|\myvec{r}-\myvec{r}_i|} + (\text{regular part}),
     \end{equation*}
     again, if we factor out the conformal factor $\psi$, the remaining conformal metric is regular everywhere.
   \end{frame}
@@ -286,6 +286,16 @@
         0 & -\myvec{\xi} & 0
       \end{pmatrix},
     \end{equation}
+    it's very clear that the principal symbol matrix is already symmetric, and thus the system is symmetrically hyperbolic, and well-posed in the $L^2$ sense.
+
+    The eigenvalues are $\lambda_1 = \lambda_2 = \lambda_3 = 0$, $\lambda_4 = -1$ and $\lambda_5 = 1$. The eigenvectors are
+    \begin{equation}
+      \myvec{e}_1 = \mqty(0\\0\\0\\-1,1), \quad
+      \myvec{e}_2 = (0,0,\xi_2,-\xi_1,0)^T, \quad
+      \myvec{e}_3 = (0,0,\xi_3,0,-\xi_1)^T, \quad
+      \myvec{e}_4 = (0,-1,\xi_1,\xi_2,\xi_3)^T, \quad
+      \myvec{e}_5 = (0,1,\xi_1,\xi_2,\xi_3)^T.
+    \end{equation}
   \end{frame}
 
   \begin{frame}{Existing first order formulations}