Matrix conditioning
Why it matters
Conditioning determines how strongly data perturbations can change the exact solution of a linear system, separating problem sensitivity from algorithm behavior.
Prerequisites
Courses 2.4 and 6.1; preceding canonical classes where applicable.
Concept and explanation
For nonsingular A, κ(A)=||A||||A⁻¹|| measures worst-case relative sensitivity. A large κ warns that small relative data changes may produce large solution changes; it does not by itself prove every computed answer is inaccurate.
Key terms
condition number; norm; residual; forward error; backward error; perturbation; scaling; singularity
Notation / representation
r=b−Ax̂. In a compatible norm, relative forward error is bounded approximately by conditioning times relative backward error under stated assumptions.
Numerical method / algorithm
Inspect structure and scale → choose norm → estimate κ → solve → compute residual → perturb data → compare solution sensitivity → report credible digits.
Mathematical development
For A=diag(1,ε) in the 2-norm, κ₂(A)=1/|ε| when 0<|ε|≤1. The second component of Ax=b amplifies relative uncertainty as ε becomes small.
Learning objectives
- Estimate and interpret matrix condition numbers in a declared norm.
- Distinguish residual, forward error, backward error and scaling effects.
Worked examples
Worked example 1
Problem · matrix/data · goal: Analyze A=diag(1,0.001) with b=(1,0.001).
Assumptions · method · algorithm · computation: κ₂(A)=1000 and x=(1,1). Perturb b₂ to 0.001001 while A is fixed.
Result · residual/error · validation · interpretation · limitations: x₂ becomes 1.001: a 0.1% change in b₂ causes a 0.1% component change; κ is a worst-case bound, not a prediction that every perturbation is maximally amplified.
Worked example 2
Problem · matrix/data · goal: A computed x̂ has ||r||₂=10⁻¹⁰ but κ₂(A)≈10⁸.
Assumptions · method · algorithm · computation: Normalize the residual and combine it with a condition estimate before judging forward accuracy.
Result · residual/error · validation · interpretation · limitations: A tiny residual supports a small backward error, but the large condition number permits much larger forward error; report both.
Common mistake and counterexample
Incorrect: a small residual guarantees an accurate solution. An ill-conditioned system can have a tiny residual and substantial forward error.
Guided practice
- Rebuild Worked Example 1 with matrix structure, dimensions, parameters and an iteration or factorization record.
- Change one input, predict sensitivity, recompute, measure residual/error and validate independently.
A/B/C/D practice
A · Foundation
Compute condition numbers for diagonal matrices in two norms.
B · Application
Assess scaling in a synthetic sensor-calibration system.
C · Reasoning and error detection
Diagnose a residual-only accuracy claim.
D · Challenge / transfer
Design perturbation evidence and a credible-digit statement.
Knowledge check and summary
- State the matrix structure, dimensions and assumptions.
- Record the algorithm, tolerance, iteration limit and stopping evidence.
- Compute a residual and interpret it together with conditioning.
- Validate with a direct method, known eigenpair, perturbation or independent implementation.
Mastery criterion: 4/4 correct with residual, conditioning/stability distinction and independent validation. Correct each miss and complete a fresh equivalent check.
Related laboratory
Next class: Iterative solvers