Eigenvalue Calculator

An eigenvector of a square matrix A is a nonzero vector v satisfying A v = lambda v. The matrix scales that direction by lambda. A positive eigenvalue preserves direction, a negative one reverses it, and zero collapses it to the zero vector. A 2x2 matrix has two eigenvalues and a 3x3 matrix has three, counting multiplicity.

It solves the characteristic polynomial det(A - lambda I) = 0. For a 2x2 the polynomial is quadratic, so the eigenvalues come from the quadratic formula (-b +/- sqrt(b^2 - 4ac)) / (2a). For a 3x3 the polynomial is cubic, and the calculator uses Cardano's method (with the trigonometric form for the three-real-root case) to get closed-form roots up to floating-point precision.

If the discriminant of the characteristic polynomial is negative, the roots come in complex conjugate pairs. Rotation matrices are the classic example: a 90-degree rotation in 2D has eigenvalues +i and -i, because no real direction is preserved. The calculator reports complex eigenvalues as real and imaginary parts and skips the eigenvector for those slots (it would also be complex).

A matrix is diagonalizable when it has a full set of linearly independent eigenvectors, enough to form an invertible matrix P with A = P D P^-1 for some diagonal D. Every symmetric matrix is diagonalizable (spectral theorem) and every matrix with n distinct eigenvalues is too. The defective case is something like [[2, 1], [0, 2]]: a repeated eigenvalue with only one independent eigenvector, so it cannot be diagonalized.

The trace equals the sum of the eigenvalues (counting multiplicity) and the determinant equals their product. A singular matrix (det = 0) always has 0 as an eigenvalue, and a matrix with trace 0 has eigenvalues that sum to zero. These are quick sanity checks: if your eigenvalues do not add to the trace or multiply to the determinant, something is off in the computation.