Eigenvalue Calculator (2x2)
Find the eigenvalues of a 2x2 matrix.
This tool is for informational and educational purposes only. It is not a substitute for professional financial, medical, legal, or engineering advice. See Terms of Service.
Can't find what you need?
Request a ToolHow to Use the Eigenvalue Calculator
This tool finds the eigenvalues of a 2x2 matrix.
- Enter the matrix. Provide the four entries a, b, c, d of the 2x2 matrix.
- View the eigenvalues. Both eigenvalues appear along with the trace, determinant, and discriminant.
- Copy or share. Use the buttons to save or share your result.
About Eigenvalues
Eigenvalues of a matrix A are scalars that satisfy det(A - lambda * I) = 0. For a 2x2 matrix, this gives a quadratic equation: lambda^2 - trace * lambda + det = 0. The trace is a + d, and the determinant is ad - bc. If the discriminant (trace^2 - 4*det) is negative, the eigenvalues are complex conjugates. Eigenvalues are fundamental in linear algebra and appear in stability analysis, principal component analysis, quantum mechanics, and many other fields.
Frequently Asked Questions
What are eigenvalues used for?
Eigenvalues are used in stability analysis, vibration problems, principal component analysis, Google's PageRank algorithm, quantum mechanics, and many other applications.
Can eigenvalues be complex numbers?
Yes. When the discriminant is negative, the eigenvalues are complex conjugates. This happens when the matrix represents a rotation or has oscillatory behavior.
How do eigenvalues relate to the trace and determinant?
The sum of the eigenvalues equals the trace, and the product of the eigenvalues equals the determinant.