For some [[Square Matrix]] $A\in M_{n\times n}$ with [[Eigenvalue|Eigenvalues]] $\set{\lambda_{i}}$, the spectral radius is defined as: $\huge \begin{align} \rho (A) &= \max_{i}|\lambda_{i}| \\ \end{align}$ The following [[Limits|limit]] [[Convergent Series|Converges]] [[Biconditional|if and only if]] $\rho(A)<1$, $ \huge \begin{align} \lim_{ k \to \infty } A^{k} = 0_{n\times n} \iff \rho(A) < 1 \end{align} $ [[Universal Quantifier|For any]] [[Matrix Norm|matrix norm]], $ \huge ||A|| \geq \rho(A) $ Let $\lambda$ be the most extreme [[Eigenvalue]] of $A$, $|\lambda | = \rho(A)$. Let $\vec x$ be an [[Eigenvector]] ($A\vec{x}=\lambda\vec x$). $\huge \begin{align} \norm{ A\vec{x} } = \norm{ \lambda \vec x } = |\lambda| \norm{ \vec{x} } \end{align} $ $ \huge \frac{\norm{ A \vec{x} } }{\norm{ \vec{x} } } = |\lambda| = \rho(A) $