Matrix Norm Calculator
Calculate 1-norm, ∞-norm, 2-norm, Frobenius norm, and max norm with step-by-step solutions
Matrix Norm Calculator
Matrix Dimensions
Norm Types
Matrix Norms
Matrix A (2×2)
1-Norm
∞-Norm
2-Norm
Frobenius Norm
Max Norm
A^T·A Trace
A^T·A (for norm calculations)
Step-by-Step Calculations
Example Calculations
2×2 Matrix Example
Matrix A:
[1 3]
Norms:
3×3 Matrix Example
Matrix A:
[1 3 9]
[6 1 0]
Norms:
Matrix Norm Types
1-Norm
||A||₁ = max column sum
Maximum absolute sum of matrix columns
∞-Norm
||A||∞ = max row sum
Maximum absolute sum of matrix rows
2-Norm
||A||₂ = √λ_max(A^T·A)
Spectral norm, induced by vector 2-norm
Frobenius
||A||F = √trace(A^T·A)
Matrix Euclidean norm
Max Norm
||A||max = max|a_ij|
Largest absolute value in matrix
Norm Properties
Non-negativity: ||A|| ≥ 0
Zero norm: ||A|| = 0 ⟺ A = 0
Homogeneity: ||cA|| = |c|·||A||
Triangle inequality: ||A+B|| ≤ ||A|| + ||B||
Submultiplicative: ||AB|| ≤ ||A||·||B||
Equivalence: All norms are equivalent
Understanding Matrix Norms
What is a Matrix Norm?
A matrix norm is a function that assigns a non-negative real number to every matrix, representing the "size" or "magnitude" of the matrix. Unlike vector norms, matrix norms often represent the maximum stretching factor when the matrix acts on unit vectors.
Mathematical Definition
Induced Matrix Norm:
Applications
- •Condition number calculations
- •Numerical stability analysis
- •Convergence analysis in iterative methods
- •Machine learning regularization
- •Signal processing and data analysis
Step 1: Choose Norm Type
Select the appropriate norm based on your application: 1-norm, ∞-norm, 2-norm, Frobenius, or max norm.
Step 2: Apply Formula
Use the specific calculation method for your chosen norm type, whether it's sums, eigenvalues, or traces.
Step 3: Interpret Result
Understand what the norm value represents in terms of matrix properties and stretching factors.