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.
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