diff --git a/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf b/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf index dc75992..654841d 100644 Binary files a/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf and b/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf differ diff --git a/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.tex b/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.tex index ff98091..861f706 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.tex @@ -4,6 +4,9 @@ \setupCheatSheet{Analysis II} +\usepackage{tikz} +\usetikzlibrary{positioning, arrows.meta} + \begin{document} \maketitle \usetcolorboxes diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex index c23549a..52734bc 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex @@ -23,3 +23,27 @@ $A$ is positive (negative) semi-definite if and only if all eigenvalues are grea (Compute Eigenvalues using $\det(A - \lambda I) = 0$) 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}