LIN-06

2×2 condition snapshot

Ill-conditioning.

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Linearcondition

Governing equation

\\kappa\\approx|\\lambda_{max}/\\lambda_{min}|

where

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

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: $\lambda_1$ , $\lambda_2$ $\rightarrow$ $\kappa$ . Ill-conditioning.

Purpose

Purpose: compute

\kappa

from

\lambda_1

\lambda_2

in Linear algebra via

\\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

\lambda_1 = 4.000\,\mathrm{—}

\lambda_2 = 0.200\,\mathrm{—}

, the governing relation

\\kappa\\approx|\\lambda_{max}/\\lambda_{min}|

yields

\kappa = 20.00\,\mathrm{—}

. Ill-conditioning. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

λ1 \lambda_14 —λ2 \lambda_20.2 —

Outputs

κ \kappa20.00 —

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Narration of this film

Ill-conditioning.

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