$\huge \begin{align*} A \vec x &= \lambda \vec x \end{align*}$ $\vec x$ is an *Eigenvector* of a square [[Matrix]] $A$, and $\lambda$ is an [[Eigenvalue]] of $A$. #### Fixed Points ($\lambda=1$) All [[Fixed Points]] of a [[Matrix]] [[Function|Transformation]] $A$ are an [[Eigenvector]] of $A$ with an [[Eigenvalue]] of $1$. $ \begin{align} A\vec x &= \vec x\\ A\vec x - \vec x &= \vec 0\\ (A-I)\vec x &= \vec 0 \\ \augmented{c|c}{A-I&\vec 0} \end{align} $ This means that [[Existential Quantifier|there exists]] a non zero fixed point of $A$ [[Biconditional|if and only if]]: - The column [[Vector|Vectors]] of $A-I$ are [[Linear Dependence|Linearly Dependent]] - $\op{rank}(A-I) < n$ - $A-I$ is *not* [[Inverse Matrices|Invertable]] - The [[Determinant]] of $A-I$ is $0$, $\det (A-I)=0$.