Eigenvalue Calculator

Calculate eigenvalues and eigenvectors with geometric interpretations and visualizations

Overview

The Eigenvalue Calculator computes eigenvalues and eigenvectors for square matrices and provides geometric interpretations through 2D visualizations. Eigenvalues and eigenvectors reveal the special directions in which a matrix transformation only scales (doesn’t rotate) - these are the fundamental building blocks for understanding how matrices transform space.

Tips

  1. Start with Diagonal Matrices: The eigenvalues of diagonal matrices are simply the diagonal entries, making them perfect for building intuition about what eigenvalues represent.

  2. Try Symmetric Matrices: Symmetric matrices always have real eigenvalues and orthogonal eigenvectors, making them easier to interpret and visualize.

  3. Use the 2D Visualization: For 2×2 matrices, the geometric visualization shows eigenvector directions and how they relate to the transformation - this visual insight is invaluable for understanding.

  4. Check the Trace and Determinant: The sum of eigenvalues equals the trace (sum of diagonal elements) and the product of eigenvalues equals the determinant - use these to verify your results.

  5. Experiment with Example Matrices: Try rotation, scaling, and reflection matrices to see how different transformations produce different eigenvalue patterns.