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linear_algebra:01_matrices [2014/01/20 11:43] marje |
linear_algebra:01_matrices [2014/12/14 00:03] jaan fixed spelling typos |
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| <BOOKMARK:m_multiplication> | <BOOKMARK:m_multiplication> | ||
| - | ^Operaton ^Formular ^Example^ | + | ^Operation ^Formula ^Example^ |
| | Sum | $(a_{ij})+(b_{ij})=(a_{ij}+b_{ij})$ | $\begin{pmatrix}\color{blue}{1} & \color{blue}2\\\color{blue}0 &\color{blue}{-2}\end{pmatrix}+\begin{pmatrix}\color{red}{0} &\color{red}{-1}\\\color{red}1 & \color{red}2\end{pmatrix}=\begin{pmatrix}\color{blue}1+\color{red}0 & \color{blue}2+\color{red}{(-1)}\\\color{blue}0+\color{red}1 & \color{blue}{-2}+\color{red}2\end{pmatrix}=\begin{pmatrix}1 & 1\\1 & 0 \end{pmatrix}$ | | | Sum | $(a_{ij})+(b_{ij})=(a_{ij}+b_{ij})$ | $\begin{pmatrix}\color{blue}{1} & \color{blue}2\\\color{blue}0 &\color{blue}{-2}\end{pmatrix}+\begin{pmatrix}\color{red}{0} &\color{red}{-1}\\\color{red}1 & \color{red}2\end{pmatrix}=\begin{pmatrix}\color{blue}1+\color{red}0 & \color{blue}2+\color{red}{(-1)}\\\color{blue}0+\color{red}1 & \color{blue}{-2}+\color{red}2\end{pmatrix}=\begin{pmatrix}1 & 1\\1 & 0 \end{pmatrix}$ | | ||
| - | | Scalar multiplication | $c\cdot (a_{ij})=(c\cdot a_{ij})$ | $\color{blue}{-1}\cdot\begin{pmatrix}1 & 8 \\5 &-2 \end{pmatrix}=\begin{pmatrix}\color{blue}{-1}\cdot 1 & \color{blue}{-1}\cdot 8 \\\color{blue}{-1}\cdot 5 &\color{blue}{-1}\cdot (-2)\end{pmatrix}=\begin{pmatrix}-1 & -8 \\-5 & 2 \end{pmatrix}$| | + | | Scalar multiplication | $c\cdot (a_{ij})=(c\cdot a_{ij})$ | $\color{blue}{-1}\cdot\begin{pmatrix}1 & 8 \\5 &-2 \end{pmatrix}=$ $\begin{pmatrix}\color{blue}{-1}\cdot 1 & \color{blue}{-1}\cdot 8 \\\color{blue}{-1}\cdot 5 &\color{blue}{-1}\cdot (-2)\end{pmatrix}=\begin{pmatrix}-1 & -8 \\-5 & 2 \end{pmatrix}$| |
| - | |Matrix multiplication | Let $AB=(c_{ij})$, then $c_{ij}=\sum^m_{r=1}a_{ir}b_{rj}$,\\ where $A=(a_{ij})$ is $n\times m$ matrix and $B=(b_{ij})$ is $m\times p$ matrix. | $\begin{pmatrix}\color{blue}2 & \color{blue}3 & \color{blue}4\\ \color{red}1 & \color{red}0 & \color{red}0\end{pmatrix}\begin{pmatrix}\color{LimeGreen}6& \color{Cyan}{10}\\\color{LimeGreen}7 & \color{Cyan}5\\\color{LimeGreen}8 & \color{Cyan}1\end{pmatrix}=\begin{pmatrix}\color{blue}2\cdot \color{LimeGreen}6+\color{blue}3\cdot \color{LimeGreen}7+\color{blue}4\cdot \color{LimeGreen}8 & \color{blue}2\cdot \color{Cyan}{10}+\color{blue}3\cdot \color{Cyan}5+\color{blue}4\cdot \color{Cyan}1\\\color{red}1\cdot \color{LimeGreen}6+\color{red}0\cdot \color{LimeGreen}7+\color{red}0\cdot \color{LimeGreen}8 & \color{red}1\cdot \color{Cyan}{10}+\color{red}0\cdot \color{Cyan}5+\color{red}0\cdot \color{Cyan}1\end{pmatrix}=\begin{pmatrix}65& 39\\6 & 10\end{pmatrix}$| | + | |Matrix multiplication | Let $AB=(c_{ij})$, then $c_{ij}=\sum^m_{r=1}a_{ir}b_{rj}$,\\ where $A=(a_{ij})$ is $n\times m$ matrix and $B=(b_{ij})$ is $m\times p$ matrix. | $\begin{pmatrix}\color{blue}2 & \color{blue}3 & \color{blue}4\\ \color{red}1 & \color{red}0 & \color{red}0\end{pmatrix}\begin{pmatrix}\color{LimeGreen}6& \color{Cyan}{10}\\\color{LimeGreen}7 & \color{Cyan}5\\\color{LimeGreen}8 & \color{Cyan}1\end{pmatrix}=$ $\begin{pmatrix}\color{blue}2\cdot \color{LimeGreen}6+\color{blue}3\cdot \color{LimeGreen}7+\color{blue}4\cdot \color{LimeGreen}8 & \color{blue}2\cdot \color{Cyan}{10}+\color{blue}3\cdot \color{Cyan}5+\color{blue}4\cdot \color{Cyan}1\\\color{red}1\cdot \color{LimeGreen}6+\color{red}0\cdot \color{LimeGreen}7+\color{red}0\cdot \color{LimeGreen}8 & \color{red}1\cdot \color{Cyan}{10}+\color{red}0\cdot \color{Cyan}5+\color{red}0\cdot \color{Cyan}1\end{pmatrix}=$ $\begin{pmatrix}65& 39\\6 & 10\end{pmatrix}$| |
| <WRAP important round> | <WRAP important round> | ||
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| * $\begin{pmatrix}2 & 10 & 7\\1 & 4 & 3\end{pmatrix}\begin{pmatrix}1 & 6 & 4\\2 & 3 & 1\\2 & 1 &10\end{pmatrix}$ over $\mathbb{Z_3}$. | * $\begin{pmatrix}2 & 10 & 7\\1 & 4 & 3\end{pmatrix}\begin{pmatrix}1 & 6 & 4\\2 & 3 & 1\\2 & 1 &10\end{pmatrix}$ over $\mathbb{Z_3}$. | ||
| - | </columns> | + | </columns>\\ |
| + | </WRAP> | ||
| <hidden **Solution**> | <hidden **Solution**> | ||
| Line 169: | Line 170: | ||
| </code> | </code> | ||
| </hidden>\\ | </hidden>\\ | ||
| - | </WRAP> | ||
| Another important operation on matrices is that of taking **the transpose**. It is denoted by placing a $T$ as an exponent on the initial matrix. | Another important operation on matrices is that of taking **the transpose**. It is denoted by placing a $T$ as an exponent on the initial matrix. | ||
| - | ^Operaton ^Formular ^Example^ | + | ^Operation ^Formula ^Example^ |
| | Transpose |$(a_{ij})^T=(a_{ji})$ | $\begin{pmatrix}\color{blue}1 & \color{blue}2 & \color{blue}{-6}\\ \color{red}3 & \color{red}5 & \color{red}4\end{pmatrix}^T=\begin{pmatrix}\color{blue}1 & \color{red}3\\ \color{blue}2 & \color{red}5\\ \color{blue}{-6} & \color{red}4\end{pmatrix}$ | | | Transpose |$(a_{ij})^T=(a_{ji})$ | $\begin{pmatrix}\color{blue}1 & \color{blue}2 & \color{blue}{-6}\\ \color{red}3 & \color{red}5 & \color{red}4\end{pmatrix}^T=\begin{pmatrix}\color{blue}1 & \color{red}3\\ \color{blue}2 & \color{red}5\\ \color{blue}{-6} & \color{red}4\end{pmatrix}$ | | ||
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| Calculate $5\cdot AA^T+\mathbf{b}\mathbf{c}$ by using IPython . | Calculate $5\cdot AA^T+\mathbf{b}\mathbf{c}$ by using IPython . | ||
| + | </WRAP> | ||
| <hidden **Solution**> | <hidden **Solution**> | ||
| Line 233: | Line 234: | ||
| </code> | </code> | ||
| </hidden>\\ | </hidden>\\ | ||
| - | </WRAP> | ||