How to use the 2x2 eigenvalue calculator
Enter the four entries of A = [[a, b], [c, d]] to get the eigenvalues λ₁ and λ₂ together with a matching eigenvector for each. The characteristic equation is λ² − (a+d)λ + (ad−bc) = 0, and this tool evaluates the discriminant as D = (a−d)² + 4bc. That is algebraically the same as (a+d)² − 4(ad−bc) but loses far fewer digits when two large numbers are subtracted.
A positive D means two distinct real eigenvalues, D = 0 means a repeated eigenvalue, and a negative D means a complex conjugate pair. In floating point D rarely lands exactly on zero, so a repeated root is declared when |D| < 1e-9. For two real eigenvalues the larger one is computed first and the other is recovered from the determinant, which avoids a cancelling subtraction.
Each eigenvector solves (A − λI)v = 0, is scaled to unit norm, and has its sign fixed so the first component is positive. When the eigenvalue is repeated and the matrix is defective there is only one independent eigenvector, so the second row is hidden with a note. Eigenvalues and eigenvectors are shown to 8 significant digits, and the check row lets you confirm that the eigenvalues sum to the trace and multiply to the determinant. Matrices of size 3x3 or larger are not supported.
Frequently Asked Questions
Solve the characteristic equation λ² − (a+d)λ + (ad−bc) = 0. This calculator first classifies the case from the sign of D = (a−d)² + 4bc, then computes the roots.
Nothing breaks. The eigenvalues are complex and are shown as a ± bi. In that case no real eigenvector exists, so the eigenvector rows are hidden.
Each vector is scaled to unit norm and its sign is fixed so the first component is positive. Any scalar multiple of an eigenvector is still an eigenvector.