From 729048ff55d65f9802a3a1bd77224d0a96ec5656 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 20:19:48 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 24 ++++--------------------
 1 file changed, 4 insertions(+), 20 deletions(-)

diff --git a/report.tex b/report.tex
index f8fbe79..1e40865 100644
--- a/report.tex
+++ b/report.tex
@@ -267,26 +267,10 @@
     \end{equation}
     write $u^I = (\phi, \pi, \tensor{\psi}{_1}, \tensor{\psi}{_2}, \tensor{\psi}{_3})$, then the system can be written in the form of~\eqref{first-order-PDE} with
     \begin{equation}
-      \tensor{A}{^1^I_J} = \begin{pmatrix}
-        0 & 0 & 0 & 0 & 0 \\
-        0 & 0 & -1 & 0 & 0 \\
-        0 & -1 & 0 & 0 & 0 \\
-        0 & 0 & 0 & 0 & 0 \\
-        0 & 0 & 0 & 0 & 0
-      \end{pmatrix} \qc
-      \tensor{A}{^2^I_J} = \begin{pmatrix}
-        0 & 0 & 0 & 0 & 0 \\
-        0 & 0 & 0 & -1 & 0 \\
-        0 & 0 & 0 & 0 & 0 \\
-        0 & -1 & 0 & 0 & 0 \\
-        0 & 0 & 0 & 0 & 0
-      \end{pmatrix} \qc
-      \tensor{A}{^3^I_J} = \begin{pmatrix}
-        0 & 0 & 0 & 0 & 0 \\
-        0 & 0 & 0 & 0 & -1 \\
-        0 & 0 & 0 & 0 & 0 \\
-        0 & 0 & 0 & 0 & 0 \\
-        0 & -1 & 0 & 0 & 0
+      \tensor{A}{^i^I_J} = \begin{pmatrix}
+        0 & 0 & 0 \\
+        0 & 0 & -\tensor{e}{_i} \\
+        0 & -\tensor{e}{^i} & 0
       \end{pmatrix} \qc
       S=0.
     \end{equation}