Eigenvalue Calculator
Find the eigenvalues and eigenvectors of a 2x2 or 3x3 matrix. This calculator solves the characteristic polynomial, computes the trace and determinant, and displays the corresponding eigenvectors.
Eigenvalues and eigenvectors are fundamental concepts in linear algebra. An eigenvector of a matrix A is a non-zero vector v such that Av = lambda * v, where lambda (the eigenvalue) is a scalar.
Finding Eigenvalues:
Solve the characteristic equation: det(A - lambda * I) = 0. For a 2x2 matrix, this gives a quadratic. For a 3x3, a cubic.
Finding Eigenvectors:
For each eigenvalue lambda, solve the system (A - lambda * I)v = 0. The solution space (null space) gives the eigenvector(s) for that eigenvalue.
Key Properties:
- The trace (sum of diagonal elements) equals the sum of eigenvalues
- The determinant equals the product of eigenvalues
- Symmetric matrices always have real eigenvalues
- Eigenvectors for distinct eigenvalues are linearly independent
Applications:
Eigenvalues and eigenvectors appear everywhere: principal component analysis (PCA) in data science, vibration analysis in engineering, Google's PageRank algorithm, quantum mechanics (observable operators), stability analysis of dynamic systems, and image compression.