Understanding Eigenvalues and Eigenvectors
Eigenvalues are fundamental concepts in linear algebra that reveal important properties about linear transformations represented by matrices. Our eigenvalue calculator helps you find these special numbers quickly for both 2×2 and 3×3 matrices, along with related matrix properties like trace and determinant.
Key Concepts in Eigenvalue Analysis:
- • Eigenvalue (λ): A scalar such that for a matrix A, there exists a non-zero vector v where Av = λv
- • Eigenvector: The vector v in the equation above that only scales when the matrix is applied
- • Characteristic Polynomial: det(A - λI) = 0, where I is the identity matrix - the roots are eigenvalues
- • Trace: Sum of diagonal elements (equals sum of eigenvalues)
- • Determinant: Product of eigenvalues, indicating whether the matrix is invertible
Real-World Applications of Eigenvalues:
- • Stability analysis in differential equations
- • Principal component analysis (PCA) in statistics
- • Quantum mechanics and Schrödinger's equation
- • Vibration analysis in mechanical engineering
- • Google's PageRank algorithm
- • Facial recognition and computer vision
- • Mechanical stress and strain analysis
- • Electrical circuit analysis and system stability
How Our Eigenvalue Calculator Works:
- For 2×2 Matrices: Direct solution from characteristic polynomial using the quadratic formula
- For 3×3 Matrices: Numerical solution using Cardano's formula for cubic equations
- Results Include: All eigenvalues (real and complex), matrix trace, and determinant
This tool is perfect for students learning linear algebra, engineers analyzing systems, or anyone needing quick eigenvalue calculations without manual computation.