What is Eigenvalues Eigenvectors Calculator and Why it Matters?
Eigenvalues and eigenvectors characterize the action of a linear operator. They appear in stability analysis, vibration modes, principal component analysis, and quantum mechanics.
Mathematical Formula and Theory
A non-zero vector v is an eigenvector of A with eigenvalue λ if A v = λ v. Equivalently, det(A − λ I) = 0. Numerical computation typically employs the QR algorithm.
How to Use This Calculator - Step by Step
- Choose matrix order (2×2 or 3×3).
- Enter the matrix entries.
- Click Compute Eigenvalues.
- Inspect the returned eigenvalues.
Solved Example with Full Calculation
Let A = [[4,1],[2,3]]. Characteristic polynomial λ² − 7λ + 10 = 0 gives λ₁ = 5, λ₂ = 2. Corresponding eigenvectors are [1,1]T and [1,−2]T.