A fixed point is an in input to some function that is equal to the input. $\huge f:x \mapsto x$ #### System for finding fixed points ([[Matrix|Matrix]] Transformations) You can get an [[Augmented Matrix]] for finding the [[Fixed Points]] of a [[Matrix|Matrix]] by doing: $\huge\begin{align*} A \vec x &= \vec x \\ A\vec x - \vec x &= \vec 0 \\ \pa{A - I}\vec x &= \vec 0 \\ &\sim \augmented{c|c}{A-I & \vec 0} \end{align*}$ >[!example]- >$\huge T: \mat{x\\y} \mapsto \mat{3x-y\\4x-y}$ > Solve for $x, y$. >$\huge\begin{align*} \vec x &= \augmented{c|c}{A-I&\vec 0} \\ &\sim \augmented{cc|c}{2&-1&0\\4&-2&0} \\ &\sim \augmented{cc|c}{2&-1&0\\0&0&0} \\ &\sim \augmented{cc|c}{1& -\frac{1}{2} &0\\0&0&0} \\ &\sim\cases{ x - \frac{1}{2}y &= 0 \\ 0 &= 0 } \end{align*}$