LIN-27

2×2 condition (diag)

Diagonal condition number.

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Governing2×2 condition (diag)

Governing equation

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

$l2 \in \left[0.05, 10\right]$

where

l1: l1 (—)
l2: l2 (—)
k: 2×2 condition (diag) (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

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

$l2 \in \left[0.05, 10\right]$

$l2 = 0.05$ $k = 160$

Along the curve

②

Domain of the variable — blue

l2_{\min} = 0.05,\quad l2_{\max} = 10

$l2 \in \left[0.05, 10\right]$

$l2 = 0.05$ $l2 = 0.05$

Along the curve

③

Response curve — green

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

$l2 \in \left[0.05, 10\right]$

$l2 = 0.05$ $k = 160$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}k}{\mathrm{d}l2}

$l2 \in \left[0.05, 10\right]$

$l2 = 0.113$ $k^{\prime} = -913.5$

Along the curve

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-27 — 2×2 condition (diag)) is the form associated with 2×2 condition (diag). Working symbols: $l1$ , $l2$ $\rightarrow$ $k$ . Diagonal condition number. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

k

from

l1

l2

in Linear algebra via

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

Diagonal condition number. 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

l1 = 8.000\,\mathrm{—}

l2 = 0.500\,\mathrm{—}

, the governing relation

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

yields

k = 16.000\,\mathrm{—}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

l1 l18 —l2 l20.5 —

Outputs

2×2 condition (diag) k16.000 —

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Diagonal condition number. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube