📐 Matrix Norm Calculator
Calculate various norms of a matrix
How to Use This Calculator
Set Matrix Dimensions
Choose rows and columns (spectral norm requires square).
Choose Norm Type
Select the type of norm you want to compute.
Enter Matrix Elements
Input all elements of your matrix.
Get Norm
View the computed norm with its description.
Formula
||A|| = different formulas for different norms
Various matrix norms measure different aspects
Frobenius Norm:
||A||_F = √(Σᵢⱼ |aᵢⱼ|²)
Square root of sum of squares of all elements
Max Norm:
||A||_∞ = maxᵢⱼ |aᵢⱼ|
Maximum absolute value among all elements
Row Sum Norm:
||A||_1 = maxᵢ Σⱼ |aᵢⱼ|
Maximum row sum of absolute values
Column Sum Norm:
||A||_∞ = maxⱼ Σᵢ |aᵢⱼ|
Maximum column sum of absolute values
Spectral Norm:
||A||_2 = √(λ_max(A^T A))
Largest singular value of A
About Matrix Norm Calculator
The Matrix Norm Calculator computes various norms of a matrix. Norms measure the "size" or "magnitude" of matrices, useful for error analysis, stability, and convergence in numerical methods.
When to Use This Calculator
- Error Analysis: Measure matrix differences
- Convergence: Check if iterations converge
- Stability: Assess numerical stability
- Optimal Control: Compute control gains
- Signal Processing: Analyze system response
Why Use Our Calculator?
- ✅ Multiple Norms: Computes 5 different norm types
- ✅ Flexible Size: Works with any matrix dimensions
- ✅ Clear Results: Shows norm value and description
- ✅ Educational: Helps understand different norms
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Norm: A measure of matrix magnitude
- Submultiplicative: ||AB|| ≤ ||A|| ||B||
- Unitarily Invariant: Spectral and Frobenius norms
- Operator Norms: Spectral norm measures transformation size
- Compatibility: Most norms are compatible with vector norms
Frequently Asked Questions
What is the difference between Frobenius and spectral norm?
Frobenius norm treats the matrix as a vector (sum of squares). Spectral norm (2-norm) is the largest singular value, measuring maximum transformation scaling. Spectral ≤ Frobenius for all matrices.
Can I compute spectral norm for non-square matrices?
Yes, spectral norm is defined for rectangular matrices too. It's the largest singular value, which can be computed via SVD. However, our calculator currently requires square matrices for simplicity.
Which norm should I use?
Depends on application: Frobenius for all-element errors, spectral for operator bounds, max for worst-case element, row/column for system analysis. Often multiple norms are computed.
Are all norms equivalent?
Yes, all matrix norms are equivalent in finite dimensions - there exist constants relating them. However, constants depend on size, so choice matters for numerical analysis.
Does norm satisfy triangle inequality?
Yes! All proper norms satisfy ||A + B|| ≤ ||A|| + ||B|| and are positive definite (||A|| > 0 if A ≠ 0) and homogeneous (||αA|| = |α| ||A||).