Matrix Product (Conventional)/Examples/Cayley's Motivation

Example of (Conventional) Matrix Product
Let there be $3$ Cartesian coordinate systems:
 * $\tuple {x, y}$, $\tuple {x', y'}$, $\tuple {x, y}$

Let them be connected by:


 * $\begin {cases} x' = x + y \\ y' = x - y \end {cases}$

and:


 * $\begin {cases} x = -x' - y' \\ y = -x' + y' \end {cases}$

The relationship between $\tuple {x, y}$ and $\tuple {x, y}$ is given by:


 * $\begin {cases}

x'' = -x' - y' = -\paren {x + y} - \paren {x - y} = -2 x \\ y'' = -x' + y' = -\paren {x + y} + \paren {x - y} = -2 y \end {cases}$

devised the compact notation that expressed the changes of coordinate systems by arranging the coefficients in an array:


 * $\begin {pmatrix} 1 & 1 \\ 1 & -1 \end {pmatrix} \begin {pmatrix} -1 & -1 \\ -1 & 1 \end {pmatrix} = \begin {pmatrix} -2 & 0 \\ 0 & -2 \end {pmatrix}$

As such, he can be considered as having invented matrix multiplication.