LIN-04

2×2 eigenvalues

Trace and det.

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Lineareigen 2

Governing equation

\\lambda=\\mathrm{tr}/2\\pm\\sqrt{(\\mathrm{tr}/2)^2-\\det}

$a \in \left[-10, 10\right]$

where

a: a (—)
b: b (—)
c: c (—)
d: d (—)
\lambda_+: λ+ (—)
\lambda_-: λ- (—)

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

\\lambda=\\mathrm{tr}/2\\pm\\sqrt{(\\mathrm{tr}/2)^2-\\det}

$a \in \left[-10, 10\right]$

$a = -10$ $\lambda_+ = 3$

Along the curve

②

Domain of the variable — blue

a_{\min} = -10,\quad a_{\max} = 10

$a \in \left[-10, 10\right]$

$a = -10$ $a = -10$

Along the curve

③

Response curve — green

\\lambda=\\mathrm{tr}/2\\pm\\sqrt{(\\mathrm{tr}/2)^2-\\det}

$a \in \left[-10, 10\right]$

$a = -10$ $\lambda_+ = 3$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\lambda_+}{\mathrm{d}a}

$a \in \left[-10, 10\right]$

$a = -9.874$ $\lambda_+^{\prime} = 0$

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-04 — 2×2 eigenvalues) is the form associated with eigen 2. Working symbols: $a$ , $b$ , $c$ , $d$ $\rightarrow$ $\lambda_+$ , $\lambda_-$ . Trace and det.

Purpose

Purpose: compute

\lambda_+

\lambda_-

from

a

b

c

d

in Linear algebra via

\\lambda=\\mathrm{tr}/2\\pm\\sqrt{(\\mathrm{tr}/2)^2-\\det}

Trace and det. 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

a = 2.000\,\mathrm{—}

b = 1.000\,\mathrm{—}

c = 0.000\,\mathrm{—}

d = 3.000\,\mathrm{—}

, the governing relation

\\lambda=\\mathrm{tr}/2\\pm\\sqrt{(\\mathrm{tr}/2)^2-\\det}

yields

\lambda_+ = 3.0000\,\mathrm{—}

\lambda_- = 2.0000\,\mathrm{—}

. Trace and det. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a2 —b b1 —c c0 —d d3 —

Outputs

λ+ \lambda_+3.0000 —
λ- \lambda_-2.0000 —

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Narration of this film

Trace and det.

Reading speed

Watch on YouTube