The norm of a matrix is a generalization of the [[Vector Norm]] for [[Linear Transformation|Linear Transformations]] described as [[Matrix|Matrices]]. Note this is distinct than the [[Vector Norm]] of a [[Vector Space]] of [[Matrix|Matrices]]. Let $M_{n \times n}$ be the [[Set|set]] of all $n \times n$ [[Square Matrix|Square Matrices]]. $||\cdot||: M_{n\times n}\to \R$ is a norm if: $\huge \begin{align} \\ ||A+B|| &\leq ||A|| + ||B|| \\ ||AB|| & \leq ||A||\, ||B|| \\ ||kA|| & \leq |k| \, ||A|| \\ ||A|| & \geq 0 \\ ||A|| = 0 & \iff A = 0_{n \times n} \end{align} $ ### Induced Matrix Norms If $||\cdot||$ is a [[Vector Norm]], then it will *induce* a [[Matrix Norm]]: $ \huge \begin{align} ||A|| = \max_{\vec x \neq 0} \frac{||A\vec x||}{||\vec x||} &= \max_{||\vec x|| = 1} ||A\vec x|| \end{align} $ For all [[Vector|Vectors]] in the [[Pre-Image]] of $A$, this defines this as the largest ratio between the [[Vector Norm|Norm]] of the [[Image]] of $\vec x$ through $A$ to the [[Vector Norm|Norm]] of $\vec x$. For the [[LP Norm|Kings Distance]] Norm, $\huge \begin{align} ||A||_{\infty} &= \max_{i}\pa{|a_{i,1}|+|a_{i,2}|+ \cdots+ |a_{i,n}|} \end{align} $