LIN-34

Scaled identity det

Diagonal of λ.

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GoverningScaled identity det

Governing equation

\\det(\\lambda I_n)=\\lambda^n

$lam \in \left[-8, 8\right]$

where

lam: lam (—)
n: n (—)
D: Scaled identity det (—)

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

\\det(\\lambda I_n)=\\lambda^n

$lam \in \left[-8, 8\right]$

$lam = -8$ $D = -512$

Along the curve

②

Domain of the variable — blue

lam_{\min} = -8,\quad lam_{\max} = 8

$lam \in \left[-8, 8\right]$

$lam = -8$ $lam = -8$

Along the curve

③

Response curve — green

\\det(\\lambda I_n)=\\lambda^n

$lam \in \left[-8, 8\right]$

$lam = -8$ $D = -512$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}D}{\mathrm{d}lam}

$lam \in \left[-8, 8\right]$

$lam = -7.899$ $D^{\prime} = 187.2$

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-34 — Scaled identity det) is the form associated with Scaled identity det. Working symbols: $lam$ , $n$ $\rightarrow$ $D$ . Diagonal of λ. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

D

from

lam

n

in Linear algebra via

\\det(\\lambda I_n)=\\lambda^n

Diagonal of λ. 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

lam = 2.000\,\mathrm{—}

n = 3.000\,\mathrm{—}

, the governing relation

\\det(\\lambda I_n)=\\lambda^n

yields

D = 8.000\,\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

lam lam2 —n n3 —

Outputs

Scaled identity det D8.000 —

Reading speed

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Narration of this film

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

Diagonal of λ. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube