Find eigenvalues via the characteristic polynomial (2×2, 3×3) or QR iteration (larger symmetric).
Eigenvalues λ satisfy det(A − λI) = 0. For 2×2 matrices the characteristic polynomial is quadratic (closed form). For 3×3 it is a cubic solved via Cardano's formula. For larger matrices, QR iteration is used — this implementation supports symmetric matrices of any size (which always have real eigenvalues). Non-symmetric matrices larger than 3×3 are not supported. Complex eigenvalues are reported when the discriminant is negative.
Characteristic Equation
det(A − λI) = 0
2×2 Formula
λ = (tr(A) ± √(tr²−4·det)) / 2
2×2 and 3×3 use exact closed-form solutions and report complex eigenvalues. For 4×4 and larger, only symmetric matrices are supported via QR iteration (all eigenvalues are real).
Yes. Real matrices can have complex eigenvalues that come in conjugate pairs. This calculator reports them for 2×2 and 3×3 matrices.