**Vector norm** $\|\cdot\|: V \mapsto \mathbb{R}_{\ge 0}$
is any non-negative function on a vector space,
that is positive for non-zero vectors, homogeneous, and satisfies the triangle inequality.
**p-norm**, $p \in [1, \infty]$, is the vector norm on the Euclidean n-space, defined by:
$\|x\|_p = (\sum_{i=1}^n |x_i|^p)^{1/p}$.
**2-norm** or **Euclidean norm** is the norm induced from the Euclidean inner product:
$\|x\|_2 = (\sum_{i=1}^n |x_i|^2)^{1/2}$.
Euclidean norm is the default vector norm,
and is often simply denoted as $\|\cdot\|$ or $|\cdot|$.

**Matrix norm** $\|\cdot\|: M_{m,n} \mapsto \mathbb{R}_{\ge 0}$
is any vector norm on a vector space of matrices that is sub-multiplicative for square matrices:
$\forall A, B \in M_n$, $\|A B\| \le \|A\| \|B\|$.
**Matrix p-norm** is the operator norm on a space of matrices
induced from the p-norm on the Euclidean spaces:
$\|A\|_p = \max_{\|x\|_p = 1} \|A x\|_p$.
**Matrix 2-norm** or **spectral norm** of a matrix is its largest singular value:
$\|A\|_2 = \sigma_1(A)$.
**Matrix 1-norm** or **maximum column sum norm** is the largest column sum:
$\|A\|_1 = \max_{j \in n} \sum_{i \in m} |a_{ij}|$.
**Matrix ∞-norm** or **maximum row sum norm** is the largest row sum:
$\|A\|_\infty = \max_{i \in m} \sum_{j \in n} |a_{ij}|$.

**L_{p,q} norm**, $p,q \in [1, \infty]$, is a function defined as:
$\|A\|_{p,q} = (\sum_{j=1}^n (\sum_{i=1}^m |a_{ij}|^p)^{q/p})^{1/q}$.
$L_{p,q}$ norm need not be a matrix norm.
$L_{p,p}$ norm of a matrix equals the p-norm of its vectorization:
$\|A\|_{p,p} = \|\text{vec}(A)\|_p$.
1-norm and 2-norms are still matrix norms, but ∞-norm is not.
**L_{2,2}-norm**, **Euclidean norm**, or **Frobenius norm** of a matrix
is the 2-norm of its vectorization:
$\|A\|_F = (\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2)^{1/2} = (\text{tr}(A A^T))^{1/2}$.
Note that the Frobenius norm of a matrix equals the Euclidean norm of its singular values:
$\|A\|_F = \|\sigma(A)\|_2$.
Therefore, the Frobenius norm is never smaller than the spectral norm: $\|A\|_F > \|A\|_2$.
**L_{1,1}-norm** of a matrix is the 1-norm of its vectorization:
$\|A\| = \sum_{i=1}^m \sum_{j=1}^n |a_{ij}|$.
**L_{2,1} norm** of a matrix is the sum of the 2-norms of its columns:
$\|A\|_{2,1} = \sum_{j=1}^n (\sum_{i=1}^m |a_{ij}|^2)^{1/2})$.
The **L_{2,1}** norm is useful if the matrix represents a data set,
where each column vector is an observation.

## Misc

Notes:

🏷 Category=Algebra Category=Matrix Theory