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[Analysis] Finish vector diff calculus

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Janis Hutz
2026-01-02 11:54:54 +01:00
parent 056524cdec
commit 47fb79d397
3 changed files with 27 additions and 0 deletions
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semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf
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semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.tex
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@@ -4,6 +4,9 @@
\setupCheatSheet{Analysis II} \setupCheatSheet{Analysis II}
\usepackage{tikz}
\usetikzlibrary{positioning, arrows.meta}
\begin{document} \begin{document}
\maketitle \maketitle
\usetcolorboxes \usetcolorboxes

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semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex
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@@ -23,3 +23,27 @@ $A$ is positive (negative) semi-definite if and only if all eigenvalues are grea
(Compute Eigenvalues using $\det(A - \lambda I) = 0$) (Compute Eigenvalues using $\det(A - \lambda I) = 0$)
For $2 \times 2$ matrices, we can use the following scheme: For $2 \times 2$ matrices, we can use the following scheme:
\begin{center}
\begin{tikzpicture}[node distance = 0.3cm and 2cm, >={Stealth[round]}]
\node (det) {$\det(A)$};
\node (indef) [below=of det] {indefinite};
\node (tr0) [right=of det] {$\text{Tr}(A)$};
\node (posdef) [above right=of tr0, yshift=-0.5cm] {pos. def.};
\node (negdef) [below right=of tr0, yshift=+0.5cm] {neg. def.};
\node (tr1) [left=of det] {$\text{Tr}(A)$};
\node (possemdef) [above left=of tr1, yshift=-0.5cm] {pos. semi-def.};
\node (negsemdef) [below left=of tr1, yshift=+0.5cm] {neg. semi-def.};
\node (zero) [below=of tr1] {$A$ is zero};
\path[->]
% Level 0
(det) edge node [above] {positive} (tr0)
(det) edge node [above] {$0$} (tr1)
(det) edge node [right] {negative} (indef)
(tr0) edge node [above] {positive} (posdef)
(tr0) edge node [below] {negative} (negdef)
(tr1) edge node [above] {positive} (possemdef)
(tr1) edge node [below] {negative} (negsemdef)
(tr1) edge node [right] {$0$} (zero);
\end{tikzpicture}
\end{center}