11.2 Matrix operations

In Module 11.1, a matrix was a labelled grid of numbers. Now we want to combine such grids. Adding matching entries combines two measurements of the same layout. Multiplying can connect two stages of a calculation: the first stage makes intermediate values, and the second stage uses them to make final values. The shapes tell us whether those intermediate values match up.

This is intermediate linear algebra. Work through OpenStax's matrix-operation examples after learning dimensions and indexing.

A basic idea: match the shapes

You may add two matrices only when they have the same shape, adding entries in matching positions. To multiply $AB$, imagine $B$ taking a vector with $p$ entries and producing $n$ intermediate values. Then $A$ turns those $n$ values into $m$ outputs. So $A$ has shape $m\times n$, $B$ has shape $n\times p$, and $AB$ has shape $m\times p$. The two inner dimensions must match because every intermediate value needs a corresponding weight.

A quick tour of formulas

For equal-shaped matrices, $(A+B)_{ij}=a_{ij}+b_{ij}$. In a product, choose the row for the output and the column for the input. Multiply entries in matching intermediate positions and add them. For a valid product,

$$(AB)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}.$$

The transpose $A^{\mathsf T}$ swaps rows and columns, so an $m\times n$ matrix becomes $n\times m$ and $(A^{\mathsf T})_{ij}=a_{ji}$. This is useful when the same data need to be grouped by columns instead of rows. In general $AB\ne BA$; reversing the stages can change the result, or even make the product undefined.

Application 1: add and transpose data grids

Two $2\times2$ measurement grids are $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B=\begin{bmatrix}5&0\\1&2\end{bmatrix}$. Their sum is

$$A+B=\begin{bmatrix}6&2\\4&6\end{bmatrix},\qquad A^{\mathsf T}=\begin{bmatrix}1&3\\2&4\end{bmatrix}.$$

Addition combines matching cells; transposition changes the orientation of the data.

Application 2: two operations in sequence

A matrix can also describe a rule that changes every point in the plane; Module 11.4 develops this view. Let $R=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ rotate a vector a quarter-turn anticlockwise, and $D=\begin{bmatrix}2&0\\0&1\end{bmatrix}$ stretch its horizontal component. Then

$$DR=\begin{bmatrix}0&-2\\1&0\end{bmatrix},\qquad RD=\begin{bmatrix}0&-1\\2&0\end{bmatrix}.$$

For $(1,0)$, $DR(1,0)=(0,1)$ while $RD(1,0)=(0,2)$. The rightmost operation acts first, so changing order changes the result.