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Undergraduate · Matrix conditioning

Matrix conditioning: A factor-of-ten inverse stretch

Matrix conditioning: investigate a factor-of-ten inverse stretch with small diagonal entry a = 0.1; final data error b = 1.

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Matrix conditioning: A factor-of-ten inverse stretch. Data perturbation: 0. Solution perturbation: 0. Condition number κ₂: 10Small data changes can be amplified01000.51solution errordata perturbation → · labeled axes rescale to this model
0<a≤1, so the matrix is invertible and its spectral condition number is exactly 1/a. Only perturbations in the second coordinate are shown. The calculations use the exact diagonal formula, not a floating-point solver benchmark.

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Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Data perturbation
0
Solution perturbation
0
Condition number κ₂
10

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Understand what you are seeing

The idea behind the motion.

A small residual or small change in data can produce a much larger solution change when an inverse strongly stretches one direction. This diagonal example isolates that mechanism without numerical roundoff. Conditioning is a property of the problem, distinct from the stability of the algorithm used to solve it. This investigation starts with Small diagonal entry a = 0.1; Final data error b = 1. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

A=diag(1,a); A⁻¹(1,δ)=(1,δ/a); κ₂(A)=1/a

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Compare the two diagonal scales of A and calculate its inverse. The starting case is “A factor-of-ten inverse stretch.”

  2. STEP 2

    Follow the changing quantity

    Increase the second data component from zero to the selected perturbation. Follow the resulting solution error δ/a.

  3. STEP 3

    Explain and test the result

    Compare the amplification with κ₂(A). Identify the direction responsible for the largest amplification and avoid claiming every perturbation attains it.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Small diagonal entry a = 0.1; Final data error b = 1. Pause the timeline at 60%. Given data perturbation = 0.6, calculate solution perturbation, condition number κ₂. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

The inverse second diagonal entry is 1/0.1=10. A data error δ=0.6 therefore becomes δ/a=0.6/0.1=6, demonstrating this direction's amplification. Results: Solution perturbation: 6; Condition number κ₂: 10. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

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Undergraduate · Algebra

Separate a small residual from a trustworthy solution

Measure how unequal matrix scales amplify input errors.