Eigenvalues & Eigenvectors

Description

Demonstrates the geometric meaning of eigenvalues and eigenvectors by applying a 2D linear transformation to the plane and contrasting how most vectors rotate and stretch while eigenvectors only stretch along their own span. The equation Av = λv is derived visually.

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Phases

#	Phase Name	Duration	Description
1	Setup	5s	Title, show NumberPlane with multiple sample vectors
2	Apply Transformation	8s	Apply matrix to all sample vectors — most rotate and change direction
3	Find Eigenvectors	8s	Highlight the two special directions that do not rotate, only scale
4	Animate Eigen Directions	8s	Show eigenvectors in gold, animate scaling along their span lines
5	Equation Display	8s	Show Av = λv, display eigenvalue equation, compute λ₁ and λ₂
6	Characteristic Polynomial	8s	Show det(A - λI) = 0, step through finding eigenvalues algebraically
7	Wrap-up	5s	Summary diagram with eigenvectors superimposed on transformed plane

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Layout

+--------------------------------------------------+
|  Title: "Eigenvalues & Eigenvectors"             |
+--------------------------------------------------+
|                                                  |
|   NumberPlane (left 65%)     | Equations (35%)  |
|                              |                  |
|   Multiple colored arrows    | Av = λv          |
|   (sample vectors)           |                  |
|                              | det(A - λI) = 0  |
|   Gold arrows = eigenvectors |                  |
|   with span lines            | λ₁, λ₂ values   |
|                              |                  |
+--------------------------------------------------+
|  Bottom: descriptive text                        |
+--------------------------------------------------+

Area Descriptions

• Left 65%: NumberPlane showing vectors before and after transformation
• Right 35%: Equation panel — Av = λv, characteristic polynomial, eigenvalue values
• Bottom: Narration text describing what is happening

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Assets & Dependencies

• Fonts: LaTeX
• Manim version: ManimCE 0.19.1

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Notes

• Matrix used: [[3, 1], [0, 2]] — eigenvalues λ₁=3, λ₂=2, easy to compute
• Sample vectors spread at various angles to contrast rotation vs eigen-scaling
• Eigenvectors displayed in GOLD (YELLOW_B), regular vectors in BLUE
• Span lines for eigenvectors: dashed lines through origin