LIN-26

Rayleigh quotient

Symmetric 2×2 Rayleigh.

Reading speed

GoverningRayleigh quotient

Governing equation

R=x^T A x/(x^T x)

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

where

a: a (—)
b: b (—)
d: d (—)
x: x (—)
y: y (—)
R: Rayleigh quotient (—)

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

R=x^T A x/(x^T x)

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

$x = -10$ $R = 2.03$

Along the curve

②

Domain of the variable — blue

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

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

$x = -10$ $x = -10$

Along the curve

③

Response curve — green

R=x^T A x/(x^T x)

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

$x = -10$ $R = 2.03$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}R}{\mathrm{d}x}

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

$x = -9.874$ $R^{\prime} = 0.006$

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-26 — Rayleigh quotient) is the form associated with Rayleigh quotient. Working symbols: $a$ , $b$ , $d$ , $x$ , $y$ $\rightarrow$ $R$ . Symmetric 2×2 Rayleigh. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

R

from

a

b

d

x

y

in Linear algebra via

R=x^T A x/(x^T x)

Symmetric 2×2 Rayleigh. 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 = 0.000\,\mathrm{—}

d = 5.000\,\mathrm{—}

x = 1.000\,\mathrm{—}

y = 1.000\,\mathrm{—}

, the governing relation

R=x^T A x/(x^T x)

yields

R = 3.500\,\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 a2 —b b0 —d d5 —x x1 —y y1 —

Outputs

Rayleigh quotient R3.500 —

Reading speed

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Narration of this film

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

Symmetric 2×2 Rayleigh. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube