Differences

This shows you the differences between two versions of the page.

--- lecture_8_slides [2017/02/16 08:52] – [Last time: row-column multiplication and matrix multiplication] rupert
+++ lecture_8_slides [2017/02/16 08:56] (current) – [Commuting matrices III] rupert
@@ Line 1: / Line 1: @@
 ~~REVEAL~~
-==== Last time: row-column multiplication and matrix multiplication ====
+==== Row-column & matrix multiplication ====
   * The **row-column product** of $a$ and $b$ is defined by \[\!\!\!\!\!\!\!\!\!\!ab=[\begin{smallmatrix}a_1&a_2&\dots&a_n\end{smallmatrix}]\left[\begin{smallmatrix}b_1\\b_2\\\vdots\\b_n\end{smallmatrix}\right]=a_1b_1+a_2b_2+\dots+a_nb_n.\]
-  * $AB=$ matrix of all possible "row-of-$A$ times col-of-$B$" products
+  * $AB=$ matrix of all "row-of-$A$ times col-of-$B$" products
-  * For example: \[\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \def\r{\left[\begin{smallmatrix}1&0&5\end{smallmatrix}\right]}\def\rr{\left[\begin{smallmatrix}2&-1&3\end{smallmatrix}\right]}\left[\begin{smallmatrix}1&0&5\\2&-1&3\end{smallmatrix}\right]\left[\begin{smallmatrix} 1&2\\3&4\\5&6\end{smallmatrix}\right]\def\s{\left[\begin{smallmatrix}1\\3\\5\end{smallmatrix}\right]}\def\ss{\left[\begin{smallmatrix}2\\4\\6\end{smallmatrix}\right]}=\left[\begin{smallmatrix}{\r\s}&{\r\ss}\\{\rr\s}&{\rr\ss}\end{smallmatrix}\right]=\left[\begin{smallmatrix}26&32\\14&18\end{smallmatrix}\right].\]
+  * \[\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \def\r{\left[\begin{smallmatrix}1&0&5\end{smallmatrix}\right]}\def\rr{\left[\begin{smallmatrix}2&-1&3\end{smallmatrix}\right]}\left[\begin{smallmatrix}1&0&5\\2&-1&3\end{smallmatrix}\right]\left[\begin{smallmatrix} 1&2\\3&4\\5&6\end{smallmatrix}\right]\def\s{\left[\begin{smallmatrix}1\\3\\5\end{smallmatrix}\right]}\def\ss{\left[\begin{smallmatrix}2\\4\\6\end{smallmatrix}\right]}=\left[\begin{smallmatrix}{\r\s}&{\r\ss}\\{\rr\s}&{\rr\ss}\end{smallmatrix}\right]=\left[\begin{smallmatrix}26&32\\14&18\end{smallmatrix}\right].\]
-==== Matrix multiplication ====
+==== Matrix multiplication: the definition ====
   * Let $A,B$ be matrices, with sizes
@@ Line 84: / Line 84: @@
   * 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!
   * See examples above.