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Diagonalization of Quadratic Forms

Diagonalization of Quadratic Forms
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2022-10-06 18:11
VectorCoordinate GeometryCalculusMatrixTransformation
This example demonstrates the diagonalization of quadratic forms through matrix transformations, showcasing geometric feature representation. Core math concepts include matrices, eigenvalues, eigenvectors, linear transformations, and quadratic forms. GeoGebra commands and objects like If, Rotate, Transpose, Vertex, Center, Textfield, Vector, Translate, Coefficients, Line, Determinant, LaTeX, Text, Element are used, with dynamic interactions to illustrate the diagonalization process. Suitable for high school and above learners to understand the connection between linear algebra and geometry, and the invariance of quadratic forms under coordinate transformations.

GeoGebra Commands & Parameters

  1. If(a ≟ c, pi / 4, (1 / 2 * atan(b / (a - c)))) → theta
  2. Rotate(e_1, theta) → Eigenvec1fixed
  3. Rotate(e_2, theta) → Eigenvec2fixed
  4. Transpose(P) → PT
  5. Vertex(g) → Vertex2, D, E, F
  6. Center(g) → C
  7. Vertex(conic) → Vertex, G, H, I
  8. Center(conic) → B
  9. Rotate(e_2, (Signo * θ)) → Eigenvec2
  10. Textfield(a) → InputBox1
  11. Textfield(b) → InputBox2
  12. Textfield(c) → InputBox3
  13. Textfield(d) → InputBox4
  14. Textfield(e) → InputBox5
  15. Textfield(f) → InputBox6
  16. If((a - c)^(2) + (2 * b^(2)) > 0, 2, If[(a - c)^(2) + (2 * b^(2)) < 0, 0, 1]) → numEigenvalues
  17. Vector(If[C ≟ C, C, Vertex2], (0, 0)) → vectorauxiliar01
  18. Translate(g, vectorauxiliar01) → g'
  19. Coefficients(g') → coefficients
  20. Vector((0, 0), If[B ≟ B, B, Vertex]) → vectorauxiliar02
  21. Rotate(e_1, (Signo * θ)) → Eigenvec1
  22. Line((0, 0), Eigenvec1) → uaxis
  23. Line((0, 0), Eigenvec2) → vaxis
  24. Translate(uaxis, Vector[(t * vectorauxiliar02)]) → uaxis'
  25. Translate(vaxis, Vector[(t * vectorauxiliar02)]) → vaxis'
  26. Center(conic) → Centre
  27. Center(g) → Centre2
  28. Determinant(A) → DetA
  29. Determinant(AQ) → DetAQ
  30. Determinant(P) → DetP
  31. LaTeX($5) → text1
  32. Text(text1, If[Δ_2 ≟ 0, Vertex + 0.5, B + 0.5], true, true) → text5
  33. If(sgn(theta) ≟ -1, abs(α), abs(α)) → limittheta
  34. Element(PTAP, 1, 1) → TestEigen1
  35. Element(PTAP, 2, 2) → TestEigen2

AI Drawing Prompt Reference

Draw the diagonalization process of a quadratic form. First, input the coefficients a, b, c, d, e, f of the quadratic form. Then, calculate the eigenvectors Eigenvec1fixed and Eigenvec2fixed based on these coefficients. Rotate the original figure using these eigenvectors to demonstrate the diagonalization process. Finally, apply transformations and rotations to show the diagonal form of the quadratic in the new coordinate system, and label the centers before and after the transformation, as well as the eigenvalues.
The prompt above was compiled from the original drawing commands and is for reference only.