2x2 Eigenvalue Calculator
Find the eigenvalues of a 2×2 matrix from its trace and determinant, handling real, repeated and complex-conjugate cases.
How it works
λ = [ trace ± √(trace² − 4·det) ] / 2, where trace = a + d and det = ad − bc
The eigenvalues of a 2×2 matrix are the roots of its characteristic equation λ² − (trace)λ + det = 0. Applying the quadratic formula gives λ = (trace ± √(trace² − 4·det)) / 2. When the discriminant is positive the eigenvalues are real and distinct; zero gives a repeated real eigenvalue; negative gives a complex-conjugate pair. Their sum always equals the trace and their product the determinant.
Worked example
For [4, 1; 2, 3]: trace = 7, det = 4·3 − 1·2 = 10, discriminant = 49 − 40 = 9. So λ = (7 ± 3)/2, giving eigenvalues 5 and 2 — and indeed 5 + 2 = 7 (the trace) and 5 · 2 = 10 (the determinant).
Frequently asked questions
What does a complex eigenvalue mean?
It signals rotational behaviour: a real matrix with complex eigenvalues rotates vectors rather than simply stretching them along fixed directions. The eigenvalues always come as a conjugate pair a ± bi.
How can I check my eigenvalues?
Add them — the sum must equal the trace (a + d). Multiply them — the product must equal the determinant (ad − bc). Both checks are shown alongside the result.
What is the discriminant used for?
It is trace² − 4·det. Its sign decides the case: positive means two distinct real eigenvalues, zero means one repeated real eigenvalue, and negative means a complex-conjugate pair.
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This calculator is for educational purposes. Double-check important results before acting on them.