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Geometry of Eigenvalues & Eigenvectors

Geometry of Eigenvalues & Eigenvectors
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2022-12-11 18:19
VectorCoordinate GeometryMatrix
In collections MATH2001 - Calculus & Linear algebra II
This example illustrates the geometric meaning of eigenvalues and eigenvectors. Key mathematical concepts include vectors and matrices. Using GeoGebra commands such as Distance, Vector, and Sequence, the example dynamically shows how eigenvalues and eigenvectors affect matrix transformations. Suitable for university-level linear algebra courses.

GeoGebra Commands & Parameters

  1. Distance(A', f) → d1
  2. Distance(A', g) → d3
  3. Distance(B', g) → d2
  4. Distance(B', f) → d4
  5. DynamicCoordinates(A', If[d1 < 0.35 ∧ h, x(A'), If[d3 < 0.35 ∧ h, x(A'), x(A')]], If[d1 < 0.35 ∧ h, f(x(A')), If[d3 < 0.35 ∧ h, g(x(A')), y(A')]]) → A
  6. Vector((0, 0), A) → u
  7. DynamicCoordinates(B', If[d2 < 0.35 ∧ h, x(B'), If[d4 < 0.35 ∧ h, x(B'), x(B')]], If[d2 < 0.35 ∧ h, g(x(B')), If[d4 < 0.35 ∧ h, f(x(B')), y(B')]]) → B
  8. Vector((0, 0), B) → v
  9. Sequence(Segment[T3 + (i * (T4 - T3) / n), T1 + (i * (T2 - T1) / n)], i, 0, n) → ver
  10. Sequence(Segment[T3 + (i * (T1 - T3) / n), T4 + (i * (T2 - T4) / n)], i, 0, n) → hor
  11. Textfield(a) → InputBox1
  12. Textfield(b) → InputBox2
  13. Textfield(c) → InputBox3
  14. Textfield(d) → InputBox4

AI Drawing Prompt Reference

Draw a graphic illustrating the geometric meaning of eigenvalues and eigenvectors. The graphic should include a transformed vector and the original vector, along with their distance relationships. Also include points, line segments, and text boxes to explain the concepts of eigenvalues and eigenvectors. The construction order should start from the origin, draw the vectors, then show the transformed vectors, and finally add points, line segments, and text boxes to explain the graphic.
The prompt above was compiled from the original drawing commands and is for reference only.