Matrix Norms and Their Definitions

Matrix Norms

• The notation for matrix norms is often expressed as follows:
    - $||A||$
    - where ( A ) represents a matrix.

• Common types of norms:
    - Frobenius Norm:
      - Defined for a matrix $A = [a_{ij}]$ as:
  $||A||<em>F = \sqrt{\sum</em>{i=1}^{m}\sum_{j=1}^{n} |a_{ij}|^2}$
      - This norm considers the square root of the sum of the squares of all the elements in the matrix.
  
    - L1 Norm:
      - Defined as the maximum absolute column sum of the matrix:
  $||A||<em>1 = \max</em>{1 \leq j \leq n} \sum_{i=1}^{m} |a_{ij}|$
      - Therefore, it calculates the largest sum of absolute values within any column.
  
    - Infinity Norm:
      - Defined as the maximum absolute row sum of the matrix:
  $||A||<em>\infty = \max</em>{1 \leq i \leq m} \sum_{j=1}^{n} |a_{ij}|$
      - This evaluates the largest sum of absolute values across any row of the matrix.