eigenvalue calculator

Use this dedicated eigenvalue calculator to analyze a 2×2 matrix quickly. It computes the characteristic polynomial, evaluates the discriminant, and identifies whether the eigenvalues are distinct real, repeated real, or complex conjugates. Accurate eigenvalue analysis helps verify system stability, understand vibration modes, and solve linear differential equations efficiently.

Matrix Input

Enter each element of your 2×2 matrix. The calculator automatically derives the trace and determinant before solving the characteristic polynomial λ² − (trace)λ + determinant = 0.

How the Eigenvalue Calculator Works

The eigenvalues λ of a 2×2 matrix A satisfy the characteristic equation λ² − (trace A)λ + det A = 0. The trace equals the sum of diagonal elements (a11 + a22), while the determinant equals a11·a22 − a12·a21. The discriminant (trace² − 4·det) determines the nature of the eigenvalues. A positive discriminant indicates two distinct real eigenvalues, zero reveals a repeated real eigenvalue, and a negative discriminant signals complex conjugate eigenvalues.

Realistic Example

Suppose a vibration model uses matrix [[6, -2], [1, 3]]. The trace equals 9 and the determinant equals 20, leading to the characteristic equation λ² − 9λ + 20 = 0. Solving gives eigenvalues λ₁ = 5 and λ₂ = 4, confirming two stable real modes. Use the inputs above (a11=6, a12=-2, a21=1, a22=3) to verify this result.