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.
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
Frequently Asked Questions
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.
Can I use the results for professional or academic purposes?
You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.
Does Eigenvalue Calculator work offline?
Once the page is loaded, the calculation logic runs entirely in your browser. If you have already opened the page, most calculators will continue to work even if your internet connection is lost, since no server requests are needed for computation.
How accurate are the results from Eigenvalue Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.