>[!quote] In a Nutshell >[[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|Function]] from a real or complex [[Vector Space|vector space]] to the non-negative real numbers that behaves in a certain, well-defined way: it commutes with scaling, obeys a form of the triangle inequality and is zero only at the origin. --- >[!info] Norm >Given a [[Vector Space|vector space]] $X$ over a field $F$, a norm of $X$ is a real-valued function $p \colon X \rightarrow \mathbb{R}$ that satisfies >1. **Triangle Inequality**$p(x+y)\leq p(x)+p(y), \quad \forall x,y \in X.$ >2. **Absolute Homogenity**$p(sx)=|s|p(x),\quad \forall x \in X, \text{ scalar }s.$ >3. **Positive Definiteness**$\text{If } p(x)=0, \text{ then } x=0,\quad \forall x \in X.$ >[!success] All Norms on finite Vector Spaces are Equivalent >Suppose that $p$ and $q$ are two norms on a vector space $X$. The norms are called equivalent, if there exists positive real constants $c$ and $C$, such that $cq(x)\leq p(x) \leq Cq(x).$This is alwys true on a finite-dimensional vector space ! However, this does not extend to infinite-dimensional vector spaces. --- >[!info] Definition - Matrix or Vector Norm >More specifically, a [[Matrices|matrix or vector]] norm is a [[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|function]] $||\cdot|| \colon \mathbb{C}^{m \times n} \rightarrow \mathbb{R}$ satisfying >1. $||\mathbf{A}+\mathbf{B}||\leq ||\mathbf{A}|| + ||\mathbf{B}||$ for all $\mathbf{A,B}\in \mathbb{C}^{m \times n}$ >2. $||\alpha \mathbf{A}|| = |\alpha| \, ||\mathbf{A}||$ for all $\alpha \in \mathbb{C}, \mathbf{A}\in \mathbb{C}^{m \times n}$ >3. $||\mathbf{A}|| \geq 0$ with equality only if $\mathbf{A}=0$ - **Vector norms** - $\ell_{p}$-norm$||x||_{p} \coloneqq (\sum\limits_{i}|x_{i}|^{p})^{\frac{1}{p}}$ - $\ell_{\infty}$-norm$||x||_{\infty} \coloneqq \max |x_{i}|$ ![[Pasted image 20240727182046.png|450|centre]] - **Different Matrix norms** - **Spectral** norm for $\mathbf{A}\in \mathbb{C}^{m \times n}$ $||\mathbf{A}||_{2} \coloneqq \max_{\mathbf{x} \neq 0}\frac{||\mathbf{Ax}||_{2}}{||\mathbf{x}||_{2}}=\max_{||\mathbf{x}||_{2} = 1}||\mathbf{Ax}||_{2}$ - Also from [[Matrices|spectral radius]] $||\mathbf{A}||_{2}=\sqrt{\rho(\mathbf{A^{*}A})}$ (square-root of largest [[Singular Value Decomposition|singular value]]) - **Frobenius** norm for $\mathbf{A}\in \mathbb{C}^{m \times n}$ $||\mathbf{A}||_{F}=\sqrt{\sum\limits_{i,j}|a_{ij}|^{2}}$ - Is equal to $\ell_{2,2}$, see below - **Tschebyschew**- / Max-Norm $\mathbf{A}\in \mathbb{C}^{m \times n}$ $||\mathbf{A}||_{\max} \coloneqq \max_{i,j}|a_{ij}|$ - $\ell_{p,q}$-mixed norms over rows $\mathbf{A}=[\mathbf{a}_1,...,\mathbf{a}_m]^\top$ $||\mathbf{A}||_{p,q}=\big(\sum\limits_{i=1}^{m}||\mathbf{a}_i||_p^q \big)^{\frac{1}{q}}.$