Eigenvalues and Eigenvectors Calculator

Eigenvectors are defined up to a nonzero scalar multiple. Any scaled version of the same direction is equally valid. The calculator reports one representative direction, but the entire line through the origin in that direction is the eigenspace.

Complex eigenvalues indicate that the transformation involves rotation in addition to scaling. For real 2x2 matrices, complex eigenvalues always come in conjugate pairs (a + bi and a - bi). The calculator reports both values and notes that eigenvectors require complex arithmetic.

For a 2x2 matrix, the characteristic equation det(A - lambda I) = 0 produces a quadratic: lambda squared - trace times lambda + determinant = 0. This is solved using the quadratic formula, giving eigenvalues in terms of trace and determinant.

Eigenvalues tell you how much a linear transformation stretches or compresses along each eigenvector direction. Eigenvalue 2 means stretching to double. Eigenvalue 0.5 means compressing to half. Eigenvalue -1 means reflecting (reversing direction).

This calculator is designed for 2x2 matrices where eigenvalues come from a quadratic equation. For 3x3 and larger, eigenvalue computation requires numerical methods. Use the dedicated eigenvalue calculator for the 2x2 case covered here.

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