Eigenvalue Calculator
Calculate eigenvalues and eigenvectors of a square matrix with step-by-step work. Enter values for instant results with step-by-step formulas.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
det(A - lambda * I) = 0
Eigenvalues are found by solving the characteristic equation, where A is the input matrix, lambda represents the eigenvalue, and I is the identity matrix. The determinant is expanded into a polynomial whose roots are the eigenvalues.
Worked Examples
Example 1: 2x2 Matrix Eigenvalues
Problem:Find the eigenvalues of A = [[4, 1], [2, 3]].
Solution:Characteristic polynomial: det(A - lambda*I) = (4-lambda)(3-lambda) - (1)(2) = lambda^2 - 7*lambda + 10\nUsing quadratic formula: lambda = (7 +/- sqrt(49-40)) / 2 = (7 +/- 3) / 2\nlambda_1 = 5, lambda_2 = 2\nEigenvector for lambda=5: (A-5I)v=0 => [-1,1; 2,-2]v=0 => v1 = (1, 1)\nEigenvector for lambda=2: (A-2I)v=0 => [2,1; 2,1]v=0 => v2 = (1, -2)
Result:Eigenvalues: 5, 2 | Trace = 7 = 5+2 | Det = 10 = 5*2
Example 2: Complex Eigenvalues
Problem:Find the eigenvalues of A = [[0, -1], [1, 0]] (rotation matrix).
Solution:Characteristic polynomial: lambda^2 + 1 = 0\nDiscriminant = 0 - 4 = -4 (negative, so complex roots)\nlambda = (0 +/- sqrt(-4)) / 2 = +/- i\nlambda_1 = i, lambda_2 = -i
Result:Eigenvalues: +i, -i | This is a 90-degree rotation matrix with purely imaginary eigenvalues
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy