Eigenvalue and Eigenvector Calculator — 2×2 Matrix

Formula

How this is calculated

An eigenvector v of a square matrix A is a non-zero vector that the matrix only scales (stretches or flips), not rotates: A·v = λ·v, where the scalar λ is the corresponding eigenvalue. To find eigenvalues, subtract λ from the diagonal and set the determinant of the result to zero: det(A − λI) = 0. For a 2×2 matrix [[a,b],[c,d]] this gives the characteristic polynomial λ² − (a+d)λ + (ad−bc) = 0, solved by the quadratic formula with trace tr = a+d and determinant det = ad−bc.

When the discriminant tr² − 4·det is positive there are two distinct real eigenvalues. For each eigenvalue λᵢ, the eigenvector is found by solving (A − λᵢI)v = 0: if b ≠ 0 the first row gives v = [b, λᵢ − a]; otherwise the second row gives v = [λᵢ − d, c]; for a diagonal matrix the standard basis vectors serve. Eigenvectors are unique only up to a non-zero scalar — any multiple of an eigenvector is also an eigenvector. When discriminant = 0 there is a repeated eigenvalue and there may be only one independent eigenvector. When discriminant < 0 the eigenvalues are complex conjugates (α ± βi) and no real eigenvectors exist.

The grid visualization shows how the 2×2 matrix deforms a unit square: the transformed area equals |det(A)|, confirming that eigenvalues relate to volume scaling. Matrices with real eigenvalues have eigenvectors pointing in directions preserved by the transformation — only the scale changes, not the direction.