Matrix Transformation: Eigenvectors and Eigenvalues

Animation Specification: Eigenvectors and Eigenvalues Visualization

Animation Description and Purpose

The animation demonstrates how a matrix transforms a unit circle into an ellipse, illustrating the concepts of eigenvectors and eigenvalues. The transformation highlights the directions (eigenvectors) in which the matrix stretches or compresses the space and the corresponding scaling factors (eigenvalues).

Mathematical Elements and Formulas

• Unit Circle: Defined by the equation $x^2 + y^2 = 1$.
• Matrix Transformation: A 2x2 matrix $A$ transforms the unit circle into an ellipse. The matrix is symmetric and positive definite to ensure real eigenvalues and orthogonal eigenvectors.
• Eigenvectors and Eigenvalues: The eigenvectors $v_1$ and $v_2$ are the principal axes of the ellipse, and the eigenvalues $\lambda_1$ and $\lambda_2$ are the lengths of the semi-major and semi-minor axes.
• Transformation Equation: For any point $(x, y)$ on the unit circle, the transformed point $(x', y')$ is given by:
  $$\begin{bmatrix} x' \\ y' \end{bmatrix} = A \begin{bmatrix} x \\ y \end{bmatrix}$$

Visual Elements

• Unit Circle: A white circle with a thin black border, centered at the origin. Radius = 1 unit.
• Transformed Ellipse: A semi-transparent blue ellipse resulting from the matrix transformation. The ellipse is filled with a gradient to distinguish it from the unit circle.
• Eigenvectors: Two arrows (red and green) representing the eigenvectors $v_1$ and $v_2$. The arrows are labeled with their corresponding eigenvalues $\lambda_1$ and $\lambda_2$.
• Grid: A light gray grid in the background for spatial reference.
• Axes: X and Y axes with ticks and labels for orientation.

Animation Sequence

1. Initial Setup (0 - 2 seconds)

   • Display the unit circle centered at the origin.
   • Show the grid and axes for context.
   • Briefly highlight the unit circle with a pulse effect to draw attention.

2. Matrix Transformation (2 - 5 seconds)

   • Gradually apply the matrix transformation to the unit circle, morphing it into an ellipse.
   • The transformation is smooth and continuous, with the ellipse forming over 3 seconds.

3. Eigenvector Introduction (5 - 8 seconds)

   • Display the eigenvectors as arrows extending from the origin to the ellipse.
   • Animate the arrows growing from the origin to their final positions on the ellipse.
   • Label the eigenvectors with their corresponding eigenvalues near the tips of the arrows.

4. Highlighting Eigenvalues (8 - 10 seconds)

   • Pulse the eigenvalues $\lambda_1$ and $\lambda_2$ to emphasize their role as scaling factors.
   • Briefly animate the ellipse stretching along the eigenvector directions to reinforce the concept.

5. Final View (10 - 12 seconds)

   • Show the final transformed ellipse with eigenvectors and eigenvalues clearly visible.
   • Fade out the grid and axes slightly to focus on the key elements.

Camera Angles and Perspectives

• The animation uses a 2D perspective with the camera positioned directly above the XY plane.
• No camera movement or rotation is used to maintain clarity and focus on the transformation.

Timing and Transitions

• Total duration: 12 seconds.
• Transitions between steps are smooth, with a slight pause (0.5 seconds) between major steps for clarity.
• The transformation from circle to ellipse is the longest segment (3 seconds) to ensure the morphing is clearly visible.

Additional Details

• The matrix $A$ is chosen to have distinct eigenvalues for clarity (e.g., $\lambda_1 = 2$ and $\lambda_2 = 0.5$).
• The eigenvectors are orthogonal and aligned with the principal axes of the ellipse.
• Text labels for eigenvalues have an opaque white background with black text for readability.

Default Assumptions

• If no specific matrix is provided, a default symmetric matrix with eigenvalues 2 and 0.5 is used.
• The unit circle and ellipse are centered at the origin for simplicity.
• The animation focuses on a single transformation for clarity, without additional distractions.