Relevant definitions used throughout Analysis II.\\
\subtext{$\textbf{A} \in \R^{m \times n},\quad x,y \in \R^n,\quad \alpha \in \R$}

\begin{footnotesize}

    \definition \textbf{Scalar Product} $x \cdot y :=\sum_{i=0}^{n} (x_i \cdot y_i)$\\
    \definition \textbf{Euclidian Norm} $||x|| := \displaystyle\sqrt{\sum_{i=1}^{n} x_i^2}$\\
    \lemma \textbf{Properties of} $||x||$
    \begin{center}
        $
        \begin{array}{ll}
            (i)     & ||x|| \geq 0 \\
            (ii)    & ||x|| \iff x = 0 \\
            (iii)   & ||\alpha x|| = \alpha \cdot ||x|| \\
            (iv)    & ||x + y|| \leq ||x|| + ||y||\quad \text{(Triangle Inequality)}
        \end{array}
        $
    \end{center}

    \definition \textbf{Definiteness}
    \begin{center}
        $
        \begin{array}{lcl}
            \text{Positive Definite} &\iffdef& x^\top \textbf{A} x > 0\ \forall x \in \R^n_{\neq 0} \\
            \text{Negative Definite} &\iffdef& x^\top \textbf{A} x < 0\ \forall x \in \R^n_{\neq 0}
        \end{array}
        $
    \end{center}
    \color{gray}\scriptsize
        If $0$ is allowed, $\textbf{A}$ is called positive/negative semi-definite.
    \color{black}\footnotesize

    \definition \textbf{Trace} $\text{Tr}(\textbf{A}) := \displaystyle\sum_{i=0}^{\text{min}(m,n)} (\textbf{A})_{i, i}$\\
    \lemma \textbf{Inverse} of $\textbf{A} \in \R^{2\times2}$
    $$
    \textbf{A}^{-1}
    =
    \begin{bmatrix}
        a & b \\
        c & d
    \end{bmatrix}^{-1}
    =
    \frac{1}{\det(\textbf{A})}
    \begin{bmatrix}
        d & -b \\
        -c & a
    \end{bmatrix}
    $$
    \color{gray}\scriptsize
        For $A \in \R^{n \times n}$: Gauss Algorithm.
    \color{black}\footnotesize
    
    \lemma \textbf{Determinant} of $\textbf{A} \in \R^{2\times2}$
    $$
        \det(\textbf{A})
        =
        \det\left(
            \begin{bmatrix}
                a & b \\
                c & d
            \end{bmatrix}
        \right) = ad - bc
    $$
    \color{gray}\scriptsize
        For $A \in \R^{n \times n}$: Cofactor method.
    \color{black}\footnotesize

    \lemma \textbf{Determinant} of $A \in \R^{3\times3}$ (Sarrus)
    % https://tex.stackexchange.com/a/32981/184539
    $$\begin{tikzpicture}[>=stealth]
        \matrix [%
            matrix of math nodes,
            column sep=1em,
            row sep=1em
        ] (sarrus) {%
            a_{11} & a_{12} & a_{13} & a_{11} & a_{12} \\
            a_{21} & a_{22} & a_{23} & a_{21} & a_{22} \\
            a_{31} & a_{32} & a_{33} & a_{31} & a_{32} \\
        };
        
        \path ($(sarrus-1-1.north west)-(0.5em,0)$) edge ($(sarrus-3-1.south west)-(0.5em,0)$)
            ($(sarrus-1-3.north east)+(0.5em,0)$) edge ($(sarrus-3-3.south east)+(0.5em,0)$)
            (sarrus-1-1)                          edge            (sarrus-2-2)
            (sarrus-2-2)                          edge[->]        (sarrus-3-3)
            (sarrus-1-2)                          edge            (sarrus-2-3)
            (sarrus-2-3)                          edge[->]        (sarrus-3-4)
            (sarrus-1-3)                          edge            (sarrus-2-4)
            (sarrus-2-4)                          edge[->]        (sarrus-3-5)
            (sarrus-3-1)                          edge[dashed]    (sarrus-2-2)
            (sarrus-2-2)                          edge[->,dashed] (sarrus-1-3)
            (sarrus-3-2)                          edge[dashed]    (sarrus-2-3)
            (sarrus-2-3)                          edge[->,dashed] (sarrus-1-4)
            (sarrus-3-3)                          edge[dashed]    (sarrus-2-4)
            (sarrus-2-4)                          edge[->,dashed] (sarrus-1-5);
        
        \foreach \c in {1,2,3} {\node[anchor=south] at (sarrus-1-\c.north) {$+$};};
        \foreach \c in {1,2,3} {\node[anchor=north] at (sarrus-3-\c.south) {$-$};};
    \end{tikzpicture}$$
    \lemma \textbf{Properties of Eigenvalues}
    $$
    \text{Tr}(\textbf{A}) = \sum_{i=0}^{n} \lambda_i \qquad\qquad \text{det}(\textbf{A}) = \prod_{i=0}^{n} \lambda_i
    $$
    \color{gray}\scriptsize
        To find $\lambda_i$ solve $\det(\textbf{A} - \lambda \textbf{I}) = 0$.
    \color{black}\footnotesize
\end{footnotesize}