Matrix Norm - Yousef's Notes

The norm of a Matrix is a non-negative number denoted by $|A|$. It is a measure of how large its elements are. $$ A = \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ a_{21} & \cdots & a_{2n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \\ \end{bmatrix} \quad A \in M_{m \times n} \text{ size of this matrix is } m \times n $$

$a_{ij}$ is the Matrix element in row $i$ and column $j$

#Types

1 Norm $|A|1 = \max{1 \leq j \leq n} \left( \sum_{i=1}^{m} |a_{ij}| \right)$ - Maximum of the sum of absolute column elements.
Infinite Norm $|A|{\infty} = \max{1 \leq i \leq m} \left( \sum_{j=1}^{n} |a_{ij}| \right)$ - Maximum of the sum of absolute row elements.
Frobenius Norm $|A|F = \sqrt{\sum{i=1}^{m} \sum_{j=1}^{n} |a_{ij}|^2}$ Square root of the sum of the square of all elements.