Matrix Norm Calculator
Calculate various matrix norms including Frobenius, 1-norm, 2-norm, and infinity norm.
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Matrix Norms
\[ \|A\|_F = \sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n} |a_{ij}|^2} \]
Square root of the sum of squared elements.
\[ \|A\|_1 = \max_{1 \leq j \leq n} \sum_{i=1}^{m} |a_{ij}| \]
Maximum absolute column sum.
\[ \|A\|_2 = \sigma_{\max}(A) \]
Largest singular value of A.
\[ \|A\|_\infty = \max_{1 \leq i \leq m} \sum_{j=1}^{n} |a_{ij}| \]
Maximum absolute row sum.
Properties:
- Non-negativity: ‖A‖ ≥ 0
- Definiteness: ‖A‖ = 0 if and only if A = 0
- Triangle inequality: ‖A + B‖ ≤ ‖A‖ + ‖B‖
- Homogeneity: ‖cA‖ = |c|‖A‖
- Submultiplicativity: ‖AB‖ ≤ ‖A‖‖B‖
Applications:
- Error analysis in numerical methods
- Convergence analysis
- Condition number calculation
- Matrix approximation
- Machine learning optimization