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Eigenvalues & Eigenvectors

Audience: UniversityCategory: Math

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 = Ξ»v is derived visually.

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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 = Ξ»v is derived visually.

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 β€” 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 = Ξ»v, display eigenvalue equation, compute λ₁ and Ξ»β‚‚
6 Characteristic Polynomial 8s Show det(A - Ξ»I) = 0, step through finding eigenvalues algebraically
7 Wrap-up 5s Summary diagram with eigenvectors superimposed on transformed plane

Layout

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

Area Descriptions

  • Left 65%: NumberPlane showing vectors before and after transformation
  • Right 35%: Equation panel β€” Av = Ξ»v, characteristic polynomial, eigenvalue values
  • Bottom: Narration text describing what is happening

Assets & Dependencies

  • Fonts: LaTeX
  • Manim version: ManimCE 0.19.1

Notes

  • Matrix used: [[3, 1], [0, 2]] β€” eigenvalues λ₁=3, Ξ»β‚‚=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