Compute eigenvalues and eigenvectors of 2x2 and 3x3 matrices. Visualize eigenvectors as invariant directions under transformation.
Enter a 2x2 or 3x3 matrix to obtain its eigenvalues and corresponding eigenvectors, then view the real eigenvector directions that remain invariant under the matrix transformation. The results connect the equation Av = lambda v with the geometry of stretching, shrinking, or reversing vectors. A common surprise is that an eigenvector is not unique: every nonzero scalar multiple represents the same direction, so a displayed vector may look different from an equally correct hand-calculated answer.
A concrete input and expected output from the current implementation.
Input
2 0 0 3
Expected output
Eigenvalues: {2, 3}. Eigenvector directions: span((1, 0)^T) for eigenvalue 2 and span((0, 1)^T) for eigenvalue 3. Multiplying (1, 0)^T by the matrix gives 2(1, 0)^T, while multiplying (0, 1)^T gives 3(0, 1)^T. The x-axis and y-axis are therefore invariant directions, scaled by factors of 2 and 3 respectively.
An eigenpair must satisfy Av = lambda v for a nonzero vector v. The eigenvalues can be checked as the roots of det(A - lambda I) = 0. For each eigenvalue, its eigenvectors form the nonzero part of the null space of A - lambda I. A repeated eigenvalue may have one or several linearly independent eigenvectors.
For a real eigenpair, the line through v remains the same line after transformation because Av is a scalar multiple of v. A positive eigenvalue preserves the vector's orientation, a negative eigenvalue reverses it, and zero collapses it to the origin. The magnitude of the eigenvalue gives the scale factor along that direction.
Both vectors are correct. If v satisfies Av = lambda v, then every nonzero multiple cv also satisfies the equation for the same eigenvalue. In particular, v and -v point along the same invariant line but have opposite arrow orientations. Normalized output can also replace your vector with a shorter or longer representative.
No. The real 90-degree rotation matrix with rows (0, -1) and (1, 0) has eigenvalues i and -i, so it has no nonzero real eigenvector and no real invariant line. Its eigenpairs exist over the complex numbers, but those complex directions cannot be represented as ordinary arrows in a real two-dimensional plot.
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