Matrix Input
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Quick Presets
Eigenvalues & Eigenvectors
Enter a square matrix and click "Calculate" to find eigenvalues and eigenvectors.
Step-by-Step Solution
Detailed computation steps will appear here.
About Eigenvalues & Eigenvectors
What are Eigenvalues and Eigenvectors?
For a square matrix A, a scalar λ is an eigenvalue and vector v is an eigenvector if: A v = λ v
Characteristic Polynomial:
Eigenvalues are roots of det(A - λI) = 0, where I is the identity matrix.
For 2×2 matrices:
If A = [[a,b],[c,d]], then λ² - (a+d)λ + (ad-bc) = 0
Eigenvalues: λ = (trace ± √(trace² - 4det)) / 2
Properties:
- Sum of eigenvalues = Trace(A)
- Product of eigenvalues = det(A)
- Symmetric matrices have real eigenvalues
- Orthogonal matrices have |λ| = 1
Methods:
- Characteristic Polynomial: Exact for 2×2, 3×3 matrices
- Power Iteration: Finds dominant (largest) eigenvalue
- QR Algorithm: Iterative method to find all eigenvalues
Applications:
- Principal Component Analysis (PCA)
- Stability analysis of differential equations
- Google PageRank algorithm
- Quantum mechanics and vibration analysis
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