Differences

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

--- lecture_8_slides [2016/02/10 15:37] – rupert
+++ lecture_8_slides [2017/02/16 08:56] (current) – [Commuting matrices III] rupert
@@ Line 1: / Line 1: @@
-==== Matrix multiplication ====
+~~REVEAL~~
+==== 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 "row-of-$A$ times col-of-$B$" products
+  * \[\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \def\r{\left[\begin{smallmatrix}1&0&5\end{smallmatrix}\right]}\def\rr{\left[\begin{smallmatrix}2&-1&3\end{smallmatrix}\right]}\left[\begin{smallmatrix}1&0&5\\2&-1&3\end{smallmatrix}\right]\left[\begin{smallmatrix} 1&2\\3&4\\5&6\end{smallmatrix}\right]\def\s{\left[\begin{smallmatrix}1\\3\\5\end{smallmatrix}\right]}\def\ss{\left[\begin{smallmatrix}2\\4\\6\end{smallmatrix}\right]}=\left[\begin{smallmatrix}{\r\s}&{\r\ss}\\{\rr\s}&{\rr\ss}\end{smallmatrix}\right]=\left[\begin{smallmatrix}26&32\\14&18\end{smallmatrix}\right].\]
+==== Matrix multiplication: the definition ====
   * Let $A,B$ be matrices, with sizes
@@ Line 23: / Line 32: @@
 ==== Example 2 ====
-If $A=\mat{1&2\\3&4\\5&6}$, $B=\mat{2&1&1\\1&2&0\\1&0&2\\1&0&2\\2&2&1}$ and $C=\mat{1&3&0&7\\0&4&6&8}$,
+If $A=\mat{1&2\\3&4\\5&6}$, $B=\mat{2&1&1\\1&2&0\\1&0&2\\2&2&1}$ and $C=\mat{1&3&0&7\\0&4&6&8}$,
   * $A$ is $3\times 2$, $B$ is $4\times 3$ and $C$ is $2\times 4$, so
   * $AB$, $CA$ and $BC$ don't exist (undefined);
@@ Line 60: / Line 69: @@
 {{page>commute}}
-  * Because it's not true in general that $AB=BA$, we say that **matrix multiplication is not commutative**.
+  * Because it's true that $AB=BA$ for every choice of matrices $A$ and $B$, we say that **matrix multiplication is not commutative**.
 ==== Commuting matrices II ====
-  * What can we say about commuting matrices?
+  * What can we say about a pair of commuting matrices?
   * Suppose $AB=BA$ and think about sizes.
     * $A$: $n\times m$
@@ Line 75: / 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.
@@ Line 95: / Line 104: @@
   - $I_nB=B=BI_n$ for any $n\times n$ matrix $B$.
     * In particular, $I_n$ commutes with every other square $n\times n$ matrix $B$.
+==== Proof that $I_nA=A$ for $A$: $n\times m$ ====
+  * $I_nA$ is $n\times m$ (from definition of matrix multiplication)
+  * So $I_nA$ has same size as $A$
+  * $\text{row}_i(I_n)=[0~0~\dots~0~1~0~\dots~0]$, with $1$ in $i$th place
+  * $\text{col}_j(A)=\mat{a_{1j}\\a_{2j}\\\vdots\\a_{nj}}$
+  * So $(i,j)$ entry of $I_nA$ is \[\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\text{row}_i(I_n)\cdot \text{col}_j(A)= 0a_{1j}+0a_{2j}+\dots+1a_{ij}+\dots+0a_{nj} =a_{ij}\]
+  * same as $(i,j)$ entry of $A$.
+  * So $I_nA=A$
+==== More proofs====
+  * Proof that $A=AI_m$ for $A$: $n\times m$ is very similar (exercise)
+  * Now if $B$ is $n\times n$, take $n=m$ and $A=B$ above:
+    * $I_nB=B$ and $BI_n=B$
+    * So $I_nB=B=BI_n$
+    * So $I_n$ commutes with $B$, for any $n\times n$ matrix $B$.