219 lines
3.7 KiB
TeX
219 lines
3.7 KiB
TeX
\section{Matrices}
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\definition{}
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A \textit{matrix} is a two-dimensional array of numbers: \\
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$$
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A =
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\begin{bmatrix}
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1 & 2 & 3 \\
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4 & 5 & 6
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\end{bmatrix}
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$$
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The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix.
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\definition{}<matvec>
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We can define the product of a matrix $A$ and a vector $v$:
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$$
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Av =
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\begin{bmatrix}
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1 & 2 & 3 \\
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4 & 5 & 6
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\end{bmatrix}
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\begin{bmatrix}
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a \\ b \\ c
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\end{bmatrix}
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=
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\begin{bmatrix}
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1a + 2b + 3c \\
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4a + 5b + 6c
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\end{bmatrix}
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$$
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Note that each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
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$$
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Av =
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\begin{bmatrix}
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\text{---} a_1 \text{---} \\
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\text{---} a_2 \text{---}
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\end{bmatrix}
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\begin{bmatrix}
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| \\
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v \\
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| \\
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\end{bmatrix}
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=
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\begin{bmatrix}
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r_1v \\
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r_2v
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\end{bmatrix}
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$$
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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.
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\problem{}
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Say you multiply a size-$m$ vector by an $m \times n$ matrix. What is the size of your result?
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\vfill
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\problem{}
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Compute the following:
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$$
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\begin{bmatrix}
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1 & 2 \\
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3 & 4 \\
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5 & 6
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\end{bmatrix}
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\begin{bmatrix}
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5 \\ 3
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\end{bmatrix}
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$$
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\vfill
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\pagebreak
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\definition{}
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We also multiply a matrix by a matrix:
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$$
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AB =
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\begin{bmatrix}
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1 & 2 \\
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3 & 4
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\end{bmatrix}
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\begin{bmatrix}
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10 & 20 \\
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100 & 200
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\end{bmatrix}
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=
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\begin{bmatrix}
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210 & 420 \\
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430 & 860
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\end{bmatrix}
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$$
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Note each element of the resulting matrix is dot product of a row of $A$ and a column of $B$:
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$$
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AB =
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\begin{bmatrix}
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\text{---} a_1 \text{---} \\
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\text{---} a_2 \text{---}
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\end{bmatrix}
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\begin{bmatrix}
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| & | \\
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v_1 & v_2 \\
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| & | \\
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\end{bmatrix}
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=
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\begin{bmatrix}
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r_1v_1 & r_1v_2 \\
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r_2v_1 & r_2vm_2 \\
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\end{bmatrix}
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$$
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\begin{center}
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\begin{tikzpicture}[>=stealth,thick,baseline]
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\begin{scope}[layer = nodes]
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\matrix[
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matrix of math nodes,
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left delimiter={[},
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right delimiter={]}
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] (A) at (0, 0){
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1 & 2 \\
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3 & 4 \\
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};
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\matrix[
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matrix of math nodes,
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left delimiter={[},
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right delimiter={]}
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] (B) at (2, 0) {
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10 & 20 \\
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100 & 200 \\
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};
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\node at (3.25, 0) {$=$};
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\matrix[
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matrix of math nodes,
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left delimiter={[},
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right delimiter={]}
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] (C) at (4.5, 0) {
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210 & 420 \\
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430 & 860 \\
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};
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\end{scope}
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\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-2mm,yshift=3mm]A-1-1) rectangle ([xshift=2mm,yshift=-3mm]A-2-1) {};
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\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-3mm,yshift=2mm]B-1-1) rectangle ([xshift=3mm,yshift=-2mm]B-1-2) {};
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\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-4mm,yshift=2mm]C-1-1) rectangle ([xshift=4mm,yshift=-2mm]C-1-1) {};
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\draw[rounded corners] ([xshift=-2mm,yshift=3mm]A-1-2) rectangle ([xshift=2mm,yshift=-3mm]A-2-2) {};
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\draw[rounded corners] ([xshift=-3mm,yshift=2mm]B-2-1) rectangle ([xshift=3mm,yshift=-2mm]B-2-2) {};
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\draw[rounded corners] ([xshift=-4mm,yshift=2mm]C-2-2) rectangle ([xshift=4mm,yshift=-2mm]C-2-2) {};
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\end{tikzpicture}
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\end{center}
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\problem{}
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Compute the following matrix product. \\
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$$
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\begin{bmatrix}
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1 & 2 \\
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3 & 4 \\
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5 & 6
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\end{bmatrix}
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\begin{bmatrix}
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9 & 8 & 7 \\
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6 & 5 & 4
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\end{bmatrix}
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$$
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\vfill
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\problem{}
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Consider the following matrix product. \\
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Compute it or explain why you can't.
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$$
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\begin{bmatrix}
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1 & 2 & 3 \\
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4 & 5 & 6
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\end{bmatrix}
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\begin{bmatrix}
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10 & 20 \\
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30 & 40
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\end{bmatrix}
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$$
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\vfill
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\problem{}
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If $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ matrix, when does the product $AB$ exist?
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\vfill
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\pagebreak
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\problem{}
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Look back to \ref{matvec}. \\
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Convince yourself that vectors are matrices. \\
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Can you multiply a matrix by a vector, as in $vA$? \\
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How does the dot prouduct relate to matrix multiplication? (transpose)
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\vfill
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\pagebreak |