Norm (Math)

Norms are a measure of a number or vector’s distance from the origin.

For Scalars §

def. Absolute Value or L1 Norm. For real or complex number $x$ its norm is simply the distance from the origin.

For Vectors in $n$-dim space §

def. Euclidian Distance or L2 Norm. For vectors, the euclidian norm of a vector $\vec{x}$ in ${\mathbb{R}}^n$ is:

$$||\vec{x}|{|}_2={\left(\sum _{\forall i}{x}_i^2\right)}^{1/2}$$

For Matricies §

Motivation. As 방무창 explained, matricies also live in vector spaces ${\mathbb{R}}^{m\times n}$. Thus it would also have a notion of “distance from origin.” The matrix norm must satisfy the following conditions for it to be reasonable:

1. $||A||\ge 0$
2. $||A||=0⟺A\text{ is all zeros}$
3. $||\alpha \cdot A||=|\alpha |\cdot ||A||$ (absolutely homogenous)
4. ! $||A||+||B||\ge ||A+B||$ (triangle ineuqality)

def. Frobeinus Norm. For a matrix $A$ of shape $n\times m$ the Frobeinus norm is:

$$||A|{|}_F:={\left(\sum _{\forall i,j}{A}_{ij}^2\right)}^{1/2}$$

Visualization. matrices - What is the difference between the Frobenius norm and the 2-norm of a matrix? - Mathematics Stack Exchange