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34 lines
1.2 KiB
TeX
34 lines
1.2 KiB
TeX
Relevant definitions used throughout Analysis II.
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\subtext{$\textbf{A} \in \R^{m \times n},\quad x,y \in \R^n,\quad \alpha \in \R$}
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\definition \textbf{Scalar Product} $x \cdot y :=\sum_{i=0}^{n} (x_i \cdot y_i)$
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\definition \textbf{Euclidian Norm} $||x|| := \displaystyle\sqrt{\sum_{i=1}^{n} x_i^2}$\\
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\subtext{Used to generalize $|x|$ in many Analysis I definitions}
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\lemma \textbf{Properties of} $||x||$
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\begin{center}
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$
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\begin{array}{ll}
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(i) & ||x|| \geq 0 \\
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(ii) & ||x|| \iff x = 0 \\
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(iii) & ||\alpha x|| = \alpha \cdot ||x|| \\
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(iv) & ||x + y|| \leq ||x|| + ||y||\quad \text{(Triangle Inequality)}
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\end{array}
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$
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\end{center}
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\definition \textbf{Definiteness}
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\begin{center}
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$
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\begin{array}{lcl}
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\text{Positive Definite} &\iffdef& x^\top \textbf{A} x > 0\ \forall x \in \R^n_{\neq 0} \\
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\text{Negative Definite} &\iffdef& x^\top \textbf{A} x < 0\ \forall x \in \R^n_{\neq 0}
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\end{array}
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$
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\end{center}
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\smalltext{If $0$ is allowed, $\textbf{A}$ is called positive/negative semi-definite.}
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\definition \textbf{Trace} $\text{Tr}(\textbf{A}) := \displaystyle\sum_{i=0}^{\text{min}(m,n)} \textbf{A}_{i, i}$
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