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Definition: The sum of the singular values of the matrix. Example: For matrix $A = \begin{bmatrix}4 & 0\\0 & 3\end{bmatrix}$, the nuclear norm is the sum of singular values, which is $4 + 3 = 7$.

Definition: The maximum absolute column sum of the matrix. Example: For matrix $A = \begin{bmatrix}1 & -2\\3 & 0\end{bmatrix}$, the 1-norm is $\max(|1| + |3|, |-2| + |0|) = 4$.

Definition: The maximum absolute row sum of the matrix. Example: For matrix $A = \begin{bmatrix}1 & -2\\3 & 0\end{bmatrix}$, the infinity norm is $\max(|1| + |-2|, |3| + |0|) = 3$.

Definition: A generalization of the nuclear norm that raises singular values to the p-th power before summing and taking the p-th root. Example: For $p=2$ and matrix $A = \begin{bmatrix}4 & 0\\0 & 3\end{bmatrix}$, the Schatten norm would be $\sqrt{4^2 + 3^2} = 5$.

Definition: A matrix norm that includes weighting factors for different elements. Example: For matrix $A = \begin{bmatrix}1 & -2\\3 & 0\end{bmatrix}$ and weights $w_{ij}$ on each element $a_{ij}$, the weighted norm could be $\sum w_{ij}|a_{ij}|$ for some choice of weights.

Definition: Generalization of the p-norm that allows for different p values. Example: If $p=q=2$ for $A = \begin{bmatrix}1 & -2\\3 & 0\end{bmatrix}$, then the Holder norm reduces to the Frobenius norm.

Definition: The largest singular value of the matrix (2-Norm). Example: For matrix $A = \begin{bmatrix}5 & 0\\0 & 2\end{bmatrix}$, the spectral norm is the largest singular value, which is 5.

Definition: The sum of the k largest singular values of the matrix. Example: For $k=1$ and matrix $A = \begin{bmatrix}6 & 0\\0 & 4\end{bmatrix}$, the Ky Fan 1-norm is the largest singular value, which is 6.

Definition: Generalization of vector norms to matrices, sum each element to the p-th power and take the p-th root. Example for p=3, For matrix $A = \begin{bmatrix}1 & -2\\3 & 0\end{bmatrix}$, compute $(|1|^3+|-2|^3+|3|^3+|0|^3)^{\frac{1}{3}} = (1+8+27+0)^{\frac{1}{3}} = 36^{\frac{1}{3}}$.

Definition: The maximum absolute row sum of the matrix. Example: For matrix $A = \begin{bmatrix}1 & 2\\-3 & 0\end{bmatrix}$, the row sum norm is $\max(|1| + |2|, |-3| + |0|) = 3$.

Definition: The largest absolute value of the elements of the matrix. Example: For matrix $A = \begin{bmatrix}1 & -3\\2 & 4\end{bmatrix}$, the maximum norm is $\max(|1|, |-3|, |2|, |4|) = 4$.

Definition: The largest singular value of the matrix. Example: For matrix $A = \begin{bmatrix}4 & 0\\0 & 3\end{bmatrix}$, the 2-norm is the larger singular value, which is 4.

Definition: The sum of the absolute values of all elements. Example: For matrix $A = \begin{bmatrix}1 & -2\\3 & 0\end{bmatrix}$, the L1-norm is $|1| + |-2| + |3| + |0| = 6$.

Definition: The maximum absolute column sum of the matrix. Example: For matrix $A = \begin{bmatrix}1 & -3\\2 & 0\end{bmatrix}$, the column sum norm is $\max(|1| + |2|, |-3| + |0|) = 4$.