From c3aeb3636a0c6b6426e2b59ffde7172eaaeaaf60 Mon Sep 17 00:00:00 2001 From: Yingjie Wang Date: Fri, 10 Apr 2026 03:19:04 -0400 Subject: [PATCH] update: auto commit --- report.tex | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/report.tex b/report.tex index 7505db0..893b70e 100644 --- a/report.tex +++ b/report.tex @@ -919,7 +919,9 @@ $} \end{frame} \begin{frame}{Appendix}{Why can't we fully automate the workflow using Mathematica?} - The correct way is to check the geometric multiplicity for each eigenvalues. Say we get eigenvalues $\lambda_1, ..., \lambda_k$ with the algebraic multiplicity $m_1, ..., m_k$. Under the assumption that all these eigenvalues are different from each other, we check $\dim\ker(A - \lambda_i I)$ for each $i$. $\dim\ker(A - \lambda_i I) \leq m_i$, so we just have to find the condition for $\dim\ker(A - \lambda_i I) \geq m_i$. $\dim\ker(A - \lambda_i I) = n - \rank(A - \lambda_i I)$, so we check $\rank(A - \lambda_i I) \leq n - m_i$. + The correct way is to check the geometric multiplicity for each eigenvalues. Say we get eigenvalues $\lambda_1, ..., \lambda_k$ with the algebraic multiplicity $m_1, ..., m_k$. Under the assumption that all these eigenvalues are different from each other, we check $\dim\ker(A - \lambda_i I)$ for each $i$. $\dim\ker(A - \lambda_i I) \leq m_i$, so we just have to find the condition for $\dim\ker(A - \lambda_i I) \geq m_i$. $\dim\ker(A - \lambda_i I) = n - \rank(A - \lambda_i I)$, so we check $\rank(A - \lambda_i I) \leq n - m_i$. The criterion is that any $(n-m_i+1)\times(n-m_i+1)$ minor of $A - \lambda_i I$ is zero. + + Nice, this is a computable condition! But physically we expect most modes have the same characteristic speed $\shift{^\xi}$ \end{frame} \end{document} \ No newline at end of file