Matrix Eigenvalues Calculator
Find eigenvalues via the characteristic polynomial (2×2, 3×3) or QR iteration (larger symmetric).
Inputs
Eigenvalues
5 2
Method Used
Characteristic polynomial (2×2 closed form)
Complex Eigenvalues?
No
Step by step
Matrix A
= 4 1 2 3
Characteristic equation
λ² − tr(A)·λ + det(A) = 0
tr(A) = 7, det(A) = 10
Eigenvalues
= 5 2
How it works
Eigenvalues λ satisfy det(A − λI) = 0. For 2×2 matrices the characteristic polynomial is quadratic (closed form). For 3×3 it is a cubic solved via Cardano's formula. For larger matrices, QR iteration is used — this implementation supports symmetric matrices of any size (which always have real eigenvalues). Non-symmetric matrices larger than 3×3 are not supported. Complex eigenvalues are reported when the discriminant is negative.
Formulas
Characteristic Equation
det(A − λI) = 0
- A
- Square matrix
- λ
- Eigenvalue
- I
- Identity matrix
2×2 Formula
λ = (tr(A) ± √(tr²−4·det)) / 2
- tr
- Trace a+d
- det
- Determinant ad−bc
Frequently Asked Questions
What matrix sizes are supported?
2×2 and 3×3 use exact closed-form solutions and report complex eigenvalues. For 4×4 and larger, only symmetric matrices are supported via QR iteration (all eigenvalues are real).
Can eigenvalues be complex?
Yes. Real matrices can have complex eigenvalues that come in conjugate pairs. This calculator reports them for 2×2 and 3×3 matrices.
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