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LIN-06

2×2 condition snapshot

Ill-conditioning.

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Linearcondition

Governing equation

kappaapprox∣lambdamax/lambdamin∣\\kappa\\approx|\\lambda_{max}/\\lambda_{min}|

λ2∈[−20,20]\lambda_2 \in \left[-20, 20\right]

where

\lambda_1
λ1 (—)
\lambda_2
λ2 (—)
\kappa
κ (—)

The figure, part by part

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LIN-06 · gauge
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①

Governing law — red

kappaapprox∣lambdamax/lambdamin∣\\kappa\\approx|\\lambda_{max}/\\lambda_{min}|

λ2∈[−20,20]\lambda_2 \in \left[-20, 20\right]

λ2=−20\lambda_2 = -20κ=0.2\kappa = 0.2

②

Domain of the variable — blue

\lambda_2_{\min} = -20,\quad \lambda_2_{\max} = 20

λ2∈[−20,20]\lambda_2 \in \left[-20, 20\right]

λ2=−20\lambda_2 = -20λ2=−20\lambda_2 = -20

③

Response curve — green

kappaapprox∣lambdamax/lambdamin∣\\kappa\\approx|\\lambda_{max}/\\lambda_{min}|

λ2∈[−20,20]\lambda_2 \in \left[-20, 20\right]

λ2=−20\lambda_2 = -20κ=0.2\kappa = 0.2

④

Rate of change — amber

dκdλ2\frac{\mathrm{d}\kappa}{\mathrm{d}\lambda_2}

λ2∈[−20,20]\lambda_2 \in \left[-20, 20\right]

λ2=−19.75\lambda_2 = -19.75κ′=0.01\kappa^{\prime} = 0.01

Lecture brief

Historical brief

Determinants, eigenvalues, least squares and Cayley–Hamilton are the nineteenth-century matrix craft behind every coupled system. The sheets invert, project and diagonalise small n. This sheet (LIN-06 — 2×2 condition snapshot) is the form associated with condition. Working symbols: λ1\lambda_1, λ2\lambda_2 →\rightarrow κ\kappa. Ill-conditioning.

Purpose

Purpose: compute κ\kappa from λ1\lambda_1, λ2\lambda_2 in Linear algebra via kappaapprox∣lambdamax/lambdamin∣\\kappa\\approx|\\lambda_{max}/\\lambda_{min}| Ill-conditioning. Use it when a real linear algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given λ1=4.000 —\lambda_1 = 4.000\,\mathrm{—}, λ2=0.200 —\lambda_2 = 0.200\,\mathrm{—}, the governing relation kappaapprox∣lambdamax/lambdamin∣\\kappa\\approx|\\lambda_{max}/\\lambda_{min}| yields κ=20.00 —\kappa = 20.00\,\mathrm{—}. Ill-conditioning. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • κ \kappa20.00 —
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Narration of this film

Ill-conditioning.

Ill-conditioning.

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