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| lecture_8_slides [2017/02/16 08:49] – rupert | lecture_8_slides [2017/02/16 08:56] (current) – [Commuting matrices III] rupert |
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| ~~REVEAL~~ | ~~REVEAL~~ |
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| ==== Matrix multiplication: example from last lecture ==== | ==== Row-column & matrix multiplication ==== |
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| * We want to define $AB$ where $A$ and $B$ are matrices of "compatible" sizes (not just rows and columns) | * The **row-column product** of $a$ and $b$ is defined by \[\!\!\!\!\!\!\!\!\!\!ab=[\begin{smallmatrix}a_1&a_2&\dots&a_n\end{smallmatrix}]\left[\begin{smallmatrix}b_1\\b_2\\\vdots\\b_n\end{smallmatrix}\right]=a_1b_1+a_2b_2+\dots+a_nb_n.\] |
| * This will generalise row-column multiplication | |
| * We build $AB$ from all possible row-column products. | |
| * For example: \[\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \def\r{\left[\begin{smallmatrix}1&0&5\end{smallmatrix}\right]}\def\rr{\left[\begin{smallmatrix}2&-1&3\end{smallmatrix}\right]}\left[\begin{smallmatrix}1&0&5\\2&-1&3\end{smallmatrix}\right]\left[\begin{smallmatrix} 1&2\\3&4\\5&6\end{smallmatrix}\right]\def\s{\left[\begin{smallmatrix}1\\3\\5\end{smallmatrix}\right]}\def\ss{\left[\begin{smallmatrix}2\\4\\6\end{smallmatrix}\right]}=\left[\begin{smallmatrix}{\r\s}&{\r\ss}\\{\rr\s}&{\rr\ss}\end{smallmatrix}\right]=\left[\begin{smallmatrix}26&32\\14&18\end{smallmatrix}\right].\] | |
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| ==== Matrix multiplication ==== | * $AB=$ matrix of all "row-of-$A$ times col-of-$B$" products |
| | * \[\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \def\r{\left[\begin{smallmatrix}1&0&5\end{smallmatrix}\right]}\def\rr{\left[\begin{smallmatrix}2&-1&3\end{smallmatrix}\right]}\left[\begin{smallmatrix}1&0&5\\2&-1&3\end{smallmatrix}\right]\left[\begin{smallmatrix} 1&2\\3&4\\5&6\end{smallmatrix}\right]\def\s{\left[\begin{smallmatrix}1\\3\\5\end{smallmatrix}\right]}\def\ss{\left[\begin{smallmatrix}2\\4\\6\end{smallmatrix}\right]}=\left[\begin{smallmatrix}{\r\s}&{\r\ss}\\{\rr\s}&{\rr\ss}\end{smallmatrix}\right]=\left[\begin{smallmatrix}26&32\\14&18\end{smallmatrix}\right].\] |
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| | ==== Matrix multiplication: the definition ==== |
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| * Let $A,B$ be matrices, with sizes | * Let $A,B$ be matrices, with sizes |
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| * If $A$ and $B$ commute, they must be square matrices of the same size. | * If $A$ and $B$ commute, they must be square matrices of the same size. |
| * **Some** square matrices $A$ and $B$ of the same size commute... | * **Some** pairs of square matrices $A$ and $B$ of the same size do commute... |
| * ....but not all! | * ....but not all! |
| * See examples above. | * See examples above. |
