This sandbox was generated by Claude (Anthropic) for the Linear Algebra 101 course materials, and is built for Module 4 (Eigenvalues and Eigenvectors). Type the four entries of a 2×2 matrix A — or pick a preset — and watch everything the module defines update at once: the trace and determinant, the characteristic polynomial λ² − tr(A) λ + det(A) = 0, its two roots the eigenvalues, and the eigenvectors as the nullspace of A − λI. Every drawn line and vector is computed — the eigen-directions and the image A v of the rotating vector all come from the arithmetic, never sketched.
An eigenvector is a direction the matrix does not turn. Rotate the unit vector v and watch its image A v swing away from it everywhere except along an eigen-line, where A v is exactly parallel to v — stretched by the eigenvalue, with no rotation. The discriminant tr(A)² − 4 det(A) sorts the three cases: two real eigenvalues, one repeated, or a complex-conjugate pair (a pure rotation-and-scale, with no real eigen-direction). The identities tr(A) = λ₁ + λ₂ and det(A) = λ₁ λ₂ are shown, live, as a check.
Notation follows the course: column vectors, m × n = rows × columns, the matrix A with entries a b / c d, tr(A) = a + d, det(A) = a d − b c, and the eigen-relation A v = λ v.
What is chosen rather than computed: the default matrix, the preset matrices (each labelled where it is selected), and the extent of the drawn grid. Nothing here is a measurement; it is arithmetic, done rather than imitated.
Course demo — linked from the Module 4 lesson deck; the page itself is English‑only for now. Set a 2×2 matrix and see its eigenvalues, its eigenvectors, and what it does to the plane. One thing to take away: the discriminant tr(A)² − 4 det(A) is negative exactly when the matrix has no real eigen-direction — and that is the same moment the matrix rotates every real vector rather than merely scaling some of them.