\section{Matrices}

\definition{}
A \textit{matrix} is a two-dimensional array of numbers: \\
$$
A =
\begin{bmatrix}
	1 & 2 & 3 \\
	4 & 5 & 6
\end{bmatrix}
$$
The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix.

\definition{}<matvec>
We can define the product of a matrix $A$ and a vector $v$:

$$
Av =
\begin{bmatrix}
	1 & 2 & 3 \\
	4 & 5 & 6
\end{bmatrix}
\begin{bmatrix}
	a \\ b \\ c
\end{bmatrix}
=
\begin{bmatrix}
	1a + 2b + 3c \\
	4a + 5b + 6c
\end{bmatrix}
$$
Note that each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:

$$
Av =
\begin{bmatrix}
	\text{---} a_1 \text{---} \\
	\text{---} a_2 \text{---}
\end{bmatrix}
\begin{bmatrix}
	| \\
	v \\
	| \\
\end{bmatrix}
=
\begin{bmatrix}
	r_1v \\
	r_2v
\end{bmatrix}
$$

Naturally, a vector can only be multiplied by a matrix if the number of rows in the vector equals the number of columns in the matrix.


\problem{}
Say you multiply a size-$m$ vector by an $m \times n$ matrix. What is the size of your result?

\vfill


\problem{}
Compute the following:

$$
\begin{bmatrix}
	1 & 2 \\
	3 & 4 \\
	5 & 6
\end{bmatrix}
\begin{bmatrix}
	5 \\ 3
\end{bmatrix}
$$

\vfill
\pagebreak

\definition{}
We also multiply a matrix by a matrix:

$$
AB =
\begin{bmatrix}
	1 & 2 \\
	3 & 4
\end{bmatrix}
\begin{bmatrix}
	10 & 20 \\
	100 & 200
\end{bmatrix}
=
\begin{bmatrix}
	210 & 420 \\
	430 & 860
\end{bmatrix}
$$
Note each element of the resulting matrix is dot product of a row of $A$ and a column of $B$:

$$
AB =
\begin{bmatrix}
	\text{---} a_1 \text{---} \\
	\text{---} a_2 \text{---}
\end{bmatrix}
\begin{bmatrix}
	|	& | \\
	v_1	& v_2 \\
	|	& | \\
\end{bmatrix}
=
\begin{bmatrix}
	r_1v_1 & r_1v_2 \\
	r_2v_1 & r_2vm_2 \\
\end{bmatrix}
$$

\begin{center}
\begin{tikzpicture}

	\begin{scope}[layer = nodes]
		\matrix[
			matrix of math nodes,
			left delimiter={[},
			right delimiter={]}
		] (A) at (0, 0){
			1 & 2 \\
			3 & 4 \\
		};

		\matrix[
			matrix of math nodes,
			left delimiter={[},
			right delimiter={]}
		] (B) at (2, 0) {
			10 & 20 \\
			100 & 200 \\
		};

		\node at (3.25, 0) {$=$};

		\matrix[
			matrix of math nodes,
			left delimiter={[},
			right delimiter={]}
		] (C) at (4.5, 0) {
			210 & 420 \\
			430 & 860 \\
		};
	\end{scope}

	\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-2mm,yshift=2mm]A-1-1) rectangle ([xshift=2mm,yshift=-2mm]A-1-2) {};

	\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-3mm,yshift=2mm]B-1-1) rectangle ([xshift=3mm,yshift=-2mm]B-2-1) {};

	\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-4mm,yshift=2mm]C-1-1) rectangle ([xshift=4mm,yshift=-2mm]C-1-1) {};


	\draw[rounded corners] ([xshift=-2mm,yshift=2mm]A-2-1) rectangle ([xshift=2mm,yshift=-2mm]A-2-2) {};

	\draw[rounded corners] ([xshift=-3mm,yshift=2mm]B-1-2) rectangle ([xshift=3mm,yshift=-2mm]B-2-2) {};

	\draw[rounded corners] ([xshift=-4mm,yshift=2mm]C-2-2) rectangle ([xshift=4mm,yshift=-2mm]C-2-2) {};
\end{tikzpicture}
\end{center}


\problem{}
Compute the following matrix product. \\

$$
\begin{bmatrix}
	1 & 2 \\
	3 & 4 \\
	5 & 6
\end{bmatrix}
\begin{bmatrix}
	9 & 8 & 7 \\
	6 & 5 & 4
\end{bmatrix}
$$

\vfill


\problem{}
Compute the following matrix product or explain why you can't.

$$
\begin{bmatrix}
	1 & 2 & 3 \\
	4 & 5 & 6
\end{bmatrix}
\begin{bmatrix}
	10 & 20 \\
	30 & 40
\end{bmatrix}
$$

\vfill


\problem{}
If $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ matrix, when does the product $AB$ exist?


\vfill
\pagebreak

\problem{}
Is matrix multiplication commutative? \\
\note{Does $AB = BA$ for all $A, B$? \\ You only need one counterexample to show this is false.}

\vfill


\definition{}
Say we have a matrix $A$. The matrix $A^T$, pronounced \say{A-transpose}, is created by turning rows of $A$ into columns, and columns into rows:

$$
\begin{bmatrix}
	1 & 2 & 3 \\
	4 & 5 & 6
\end{bmatrix} ^ T
=
\begin{bmatrix}
	1 & 4 \\
	2 & 5 \\
	3 & 6
\end{bmatrix}
$$

\problem{}
Compute the following:

\hfill
$
\begin{bmatrix}
	a & b \\
	c & d
\end{bmatrix} ^ T
$\hfill
$
\begin{bmatrix}
	1 \\
	3 \\
	3 \\
	7 \\
\end{bmatrix} ^ T
$\hfill
$
\begin{bmatrix}
	1 & 2 & 4 & 8 \\
\end{bmatrix} ^ T
$
\hfill~

\vfill
\pagebreak

The \say{transpose} operator is often used to write column vectors compactly. \\
Vertical arrays don't look good in horizontal text.

\problem{}
Consider the vectors $a = [1, 2, 3]^T$ and $b = [40, 50, 60]^T$ \\
\begin{itemize}
	\item Compute the dot product $ab$.
	\item Can you redefine the dot product using matrix multiplication?
\end{itemize}
\note{As you may have noticed, a vector is a special case of a matrix.}

\vfill

\problem{}
A \textit{column vector} is an $m \times 1$ matrix. \\
A \textit{row vector} is a $1 \times m$ matrix. \\
We usually use column vectors. Why? \\
\hint{How does vector-matrix multiplication work?}


\vfill
\pagebreak