eth-summaries/semester3/analysis-ii/cheat-sheet-rb/parts/01_linalg.tex

Relevant definitions used throughout Analysis II.

\subtext{$\textbf{A} \in \R^{m \times n},\quad x,y \in \R^n,\quad \alpha \in \R$}

\definition \textbf{Scalar Product} $x \cdot y :=\sum_{i=0}^{n} (x_i \cdot y_i)$

\definition \textbf{Euclidian Norm} $||x|| := \displaystyle\sqrt{\sum_{i=1}^{n} x_i^2}$\\
\subtext{Used to generalize $|x|$ in many Analysis I definitions}

\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}
\smalltext{If $0$ is allowed, $\textbf{A}$ is called positive/negative semi-definite.}

\definition \textbf{Trace} $\text{Tr}(\textbf{A}) := \displaystyle\sum_{i=0}^{\text{min}(m,n)} (\textbf{A})_{i, i}$

\begin{footnotesize}

    \lemma \textbf{Determinant} of $\textbf{A} \in \R^{2\times2}$
    $$
        \det(\textbf{A})
        =
        \det\left(
            \begin{bmatrix}
                a & b \\
                c & d
            \end{bmatrix}
        \right) = ad - bc
    $$

    \lemma \textbf{Inverse} of $\textbf{A} \in \R^{2\times2}$
    $$
    \textbf{A}^{-1}
    =
    \begin{bmatrix}
        a & b \\
        c & d
    \end{bmatrix}^{-1}
    =
    \frac{1}{\det(\textbf{A})}
    \begin{bmatrix}
        d & -b \\
        -c & a
    \end{bmatrix}
    $$
\end{footnotesize}