2x2 Eigenvalue Calculator
Compute the eigenvalues of any 2×2 matrix by solving its characteristic equation, returning real or complex-conjugate roots.
Second eigenvalue λ₂: 2
- 1
Trace T = a + d
4 + 3 = 7 - 2
Determinant D = ad − bc
4 × 3 − 1 × 2 = 10 - 3
Discriminant Δ = T² − 4D
7² − 4 × 10 = 9 - 4
Eigenvalue λ₁ = (T + √Δ) ÷ 2
(7 + 3) ÷ 2 = 5
How does this calculator work?
For a 2×2 matrix A = [[a,b],[c,d]], the eigenvalues solve λ² − Tλ + D = 0 with trace T = a+d and determinant D = ad−bc. Using the quadratic formula, λ = (T ± √(T²−4D))/2. A non-negative discriminant gives real roots; a negative one gives a complex-conjugate pair.
Formula
How this is calculated
Enter the 2×2 matrix A = [[a, b], [c, d]] as two rows of two numbers. The calculator reads all four entries, then computes the trace T = a + d and the determinant D = ad − bc. These two scalars fully determine the eigenvalues of a 2×2 matrix.
The eigenvalues are the roots of the characteristic polynomial det(A − λI) = λ² − Tλ + D = 0. Solving with the quadratic formula gives λ = (T ± √Δ)/2 where the discriminant Δ = T² − 4D. When Δ ≥ 0 the two eigenvalues are real (equal when Δ = 0); when Δ < 0 they form a complex-conjugate pair λ = T/2 ± (√(−Δ)/2)·i.
All entries are treated as exact real numbers and results are rounded to four significant places for display. The method works for any real 2×2 matrix, including singular matrices (D = 0, giving an eigenvalue of 0) and repeated-root cases. Note that eigenvalues describe scaling directions of the linear map; the matching eigenvectors are not computed here.
Frequently asked questions
When the discriminant Δ = T² − 4D is negative. Then the roots are a complex-conjugate pair T/2 ± (√(−Δ)/2)·i, which commonly indicates rotational behavior in the linear map.
The trace equals the sum of the eigenvalues (λ₁ + λ₂ = T) and the determinant equals their product (λ₁·λ₂ = D). These relations are a quick way to check the computed values.
Yes. When Δ = 0 the matrix has a repeated eigenvalue λ = T/2. The matrix may or may not be diagonalizable in that case, but the single eigenvalue value is correct.
Also known as
TG we-Calculate Editorial Team. (2026). 2x2 Eigenvalue Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/eigenvalue-calculator-2x2
TG we-Calculate Editorial Team. "2x2 Eigenvalue Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/eigenvalue-calculator-2x2.
TG we-Calculate Editorial Team, "2x2 Eigenvalue Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/eigenvalue-calculator-2x2
@misc{wecalculate_eigenvalue_calculator_2x2, title = {2x2 Eigenvalue Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/eigenvalue-calculator-2x2}}, year = {2026}, note = {TG we-Calculate} }
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