LIN-22

Frobenius 2×2

Entrywise 2-norm.

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GoverningFrobenius 2×2

Governing equation

\\|A\\|_F=\\sqrt{a^2+b^2+c^2+d^2}

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

where

a: a (—)
b: b (—)
c: c (—)
d: d (—)
F: Frobenius 2×2 (—)

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

\\|A\\|_F=\\sqrt{a^2+b^2+c^2+d^2}

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

$a = -10$ $F = 10.95$

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

\\|A\\|_F=\\sqrt{a^2+b^2+c^2+d^2}

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

$a = -10$ $F = 10.95$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}F}{\mathrm{d}a}

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

$a = -9.874$ $F^{\prime} = -0.911$

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-22 — Frobenius 2×2) is the form associated with Frobenius 2×2. Working symbols: $a$ , $b$ , $c$ , $d$ $\rightarrow$ $F$ . Entrywise 2-norm. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

F

from

a

b

c

d

in Linear algebra via

\\|A\\|_F=\\sqrt{a^2+b^2+c^2+d^2}

Entrywise 2-norm. 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 = 1.000\,\mathrm{—}

b = 2.000\,\mathrm{—}

c = 0.000\,\mathrm{—}

d = 4.000\,\mathrm{—}

, the governing relation

\\|A\\|_F=\\sqrt{a^2+b^2+c^2+d^2}

yields

F = 4.583\,\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

a a1 —b b2 —c c0 —d d4 —

Outputs

Frobenius 2×2 F4.583 —

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Narration of this film

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

Entrywise 2-norm. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube