2×2 Matrix Eigenvalues
Roots of λ² − tr(M)λ + det(M) = 0.
Result
Eigenvalues
λ₁ = 5.000000, λ₂ = 2.000000
Two real eigenvalues.
- Matrix[[4,2],[1,3]]
- Trace tr(M)7.0000
- det(M)10.0000
- Discriminant9.0000
- λ₁5.000000
- λ₂2.000000
Step-by-step
- Characteristic polynomial: λ² − tr λ + det = 0.
- tr = a+d = 7; det = ad−bc = 10.0000.
- Discriminant = tr² − 4·det = 9.0000.
- λ = (tr ± √disc) / 2.
How to use this calculator
- Enter all 4 entries.
About this calculator
Eigenvalues of 2×2 matrix: roots of characteristic polynomial λ² − tr(M)λ + det(M) = 0. λ = (tr ± √(tr²−4det)) / 2. Real eigenvalues: matrix has real eigenvectors (basis change to diagonal form). Complex pair: rotation matrix. Used in stability analysis (linear ODEs), PCA (principal components), Markov chains, quantum mechanics. Source: NIST DLMF; Wolfram MathWorld - Eigenvalue.
Frequently asked
Complex conjugate eigenvalues. Matrix represents rotation (if real-valued).
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