Condition Number Calculator
Calculate matrix condition number to assess numerical stability and error sensitivity
Matrix Input & Configuration
Quick Examples
Condition Number Analysis
Interpretation
Condition Number Guide
Well-Conditioned (< 10)
Stable numerical computations, low error sensitivity.
Moderate (10-100)
Some sensitivity to input errors, generally acceptable.
Poor (100-1000)
High error sensitivity, use caution in computations.
Ill-Conditioned (> 1000)
Very unstable, results may be unreliable.
Matrix Norms
||A||₁ = max_j Σᵢ |aᵢⱼ|
1-norm (column sum)
||A||₂ = σ_max(A)
2-norm (spectral)
||A||∞ = max_i Σⱼ |aᵢⱼ|
∞-norm (row sum)
Condition Number Formula
cond(A) = ||A|| × ||A⁻¹||
Product of matrix and inverse norms
• If det(A) = 0, then cond(A) = ∞
• Minimum value: cond(A) = 1
• Identity matrix: cond(I) = 1
• Error bound: ||δx||/||x|| ≤ cond(A) × ||δb||/||b||
Understanding Condition Numbers
What is a Condition Number?
The condition number of a matrix measures how sensitive the solution of a linear system is to small changes in the input. It's defined as cond(A) = ||A|| × ||A⁻¹||, where ||·|| represents a matrix norm.
Physical Interpretation
- •Measures the ratio of maximum stretching to maximum shrinking
- •Indicates how "close" a matrix is to being singular
- •Determines error magnification in linear systems
Applications
Condition numbers are crucial in numerical analysis for assessing the reliability of computational results and predicting how errors propagate through calculations.
Linear Systems
For Ax = b, the condition number determines how errors in b affect the solution x.
Matrix Inversion
High condition numbers indicate numerical instability in matrix inversion.
Least Squares
Affects the reliability of regression and curve fitting solutions.
Example: Error Magnification
Problem Setup
[7.8 1.9]
b = [7.4]
[15.6]
Solution: x = [2, 0]ᵀ
With Small Error
[15.6]
x_err = [3.9]
[-7.8]
Tiny change in b causes huge change in x!
Result: With cond(A) ≈ 7,895, a 0.13% error in b produces a 400% error in x.