107 lines
3.5 KiB
TeX
107 lines
3.5 KiB
TeX
\documentclass[../../script.tex]{subfiles}
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% !TEX root = ../../script.tex
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\begin{document}
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\section{Matrices and Gaussian elimination}
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\begin{defi}
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Let $a_{ij} \in \field$, with $i \in \set{1, \cdots, n}$, $j \in \set{1, \cdots, m}$. Then
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\[
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\begin{pmatrix}
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a_{11} & a_{12} & \cdots & a_{1m} \\
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a_{21} & a_{22} & \cdots & a_{2m} \\
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\vdots & \vdots & \ddots & \vdots \\
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a_{n1} & a_{n2} & \cdots & a_{nm}
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\end{pmatrix}
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\]
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is called an $n \times m$-matrix. $(n, m)$ is said to be the dimension of the matrix. An alternative notation is
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\[
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A = (a_{ij}) \in \field^{n \times m}
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\]
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$\field^{n\times m}$ is the space of all $n \times m$-matrices. The following operations are defined for $A, B \in \field^{n \times m}$, $C \in \field^{m \times l}$:
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\begin{enumerate}[(i)]
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\item Addition
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\[
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A + B =
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\begin{pmatrix}
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a_{11} + b_{11} & \cdots & a_{1m} + b_{1m} \\
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\vdots & \ddots & \vdots \\
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a_{n1} + b_{n1} & \cdots & a_{nm} + b_{nm}
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\end{pmatrix}
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\]
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\item Scalar multiplication
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\[
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\alpha \cdot A =
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\begin{pmatrix}
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\alpha a_{11} & \cdots & \alpha a_{1m} \\
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\vdots & \ddots & \vdots \\
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\alpha a_{n1} & \cdots & \alpha a_{nm}
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\end{pmatrix}
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\]
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\item Matrix multiplication
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\[
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A \cdot C =
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\begin{pmatrix}
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a_{11}c_{11}+a_{12}c_{21}+\cdots+a_{1m}c_{m1} & \cdots & a_{11}c_{1l}+a_{12}c_{2l}+\cdots+a_{1m}c_{ml} \\
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\vdots & \ddots & \vdots \\
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a_{n1}c_{11}+a_{n2}c_{21}+\cdots+a_{nm}c_{m1} & \cdots & a_{n1}c_{1l}+a_{n2}c_{2l}+\cdots+a_{nm}c_{ml}
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\end{pmatrix}
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\]
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or in shorthand notation
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\[
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(AC)_{ij} = \series[m]{k} a_{ik}c_{kj}
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\]
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\item Transposition
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The transposed matrix $A^T \in \field^{m \times n}$ is created by writing the rows of $A$ as the columns of $A^T$ (and vice versa).
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\item Conjugate transposition
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\[
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\conj{A} = \left(\overline{A}\right)^T
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\]
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\end{enumerate}
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\end{defi}
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\begin{rem}\leavevmode
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\begin{enumerate}[(i)]
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\item $\field^{n \times m}$ (for $n, m \in \natn$) is a vector space.
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\item $A \cdot B$ is only defined if $A$ has as many columns as $B$ has rows.
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\item $\field^{n \times 1}$ and $\field^{1 \times n}$ can be trivially identified with $\field^n$.
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\item Let $A, B, C, D, E$ matrices of fitting dimensions and $\alpha \in \field$. Then
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\begin{align*}
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(A + B) C &= AC + BC \\
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A(B + C) &= AB + AC \\
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A(CE) &= (AC)E \\
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\alpha (AC) &= (\alpha A) C = A (\alpha C)
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\end{align*}
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\begin{align*}
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(A + B)^T &= A^T + B^T & \conj{(A + B)} &= \conj{A} + \conj{B} \\
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(\alpha A)^T &= \alpha (A)^T & \conj{(\alpha A)} &= \overline{A} \conj{A} \\
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(AC)^T &= C^T \cdot A^T & \conj{(AC)} &= \conj{C} \conj{A}
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\end{align*}
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\begin{proof}[Proof of associativity]
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Let $A \in \field^{n \times m}, C \in \field^{m \times l}, E \in \field^{l \times p}$. Furthermore let $i \in \set{1, \cdots, n}, j \in \set{1, \cdots, p}$.
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\begin{equation}
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\begin{split}
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\left((AC)E\right)_{ij} &= \sum_{k=1}^l (AC)_{ik} E_{kj} = \sum_{k=1}^l \left(\sum_{\tilde{k} = 1}^m a_{i\tilde{k}} c_{\tilde{k}k}\right) \cdot e_{kj} \\
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&= \sum_{k=1}^l \sum_{\tilde{k} = 1}^m a_{i\tilde{k}} \cdot c_{\tilde{k}k} \cdot e_{kj} \\
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&= \sum_{\tilde{k} = 1}^m a_{i\tilde{k}} \left( \sum_{k=1}^l c_{\tilde{k} k} e_{kj}\right) \\
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&= \sum_{\tilde{k} = 1}^m a_{i \tilde{k}} \cdot (CE)_{\tilde{k}j} \\
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&= (A(CE))_{ij}
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\end{split}
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\end{equation}
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\begin{equation}
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\implies A(CE) = A(CE)
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\end{equation}
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\end{proof}
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\end{enumerate}
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\end{rem}
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\end{document} |