Eigenvalue Calculator
Eigenvalues are the roots of the characteristic equation det(A − λI) = 0: for [[4, 1], [2, 3]] they are λ₁ = 5, λ₂ = 2.
A 2 × 2 or 3 × 3 matrix; separate entries with spaces or commas.
λ₁ = 5, λ₂ = 2
- Trace (sum of eigenvalues)
- 7
- Determinant (product of eigenvalues)
- 10
- Characteristic equation
- λ² − (7)λ + (10) = 0
Step-by-step solution
- Characteristic equation det(A − λI) = 0: λ² − (7)λ + (10) = 0
- Roots: λ₁ = 5, λ₂ = 2
- Check: sum = trace = 7, product = det A = 10
Eigenvalue Formula
det(A − λI) = 0
Σλ = trace A, Πλ = det A
Worked Example
The matrix [[4, 1], [2, 3]]:
- Characteristic equation det(A − λI) = 0: λ² − (7)λ + (10) = 0
- Roots: λ₁ = 5, λ₂ = 2
- Check: sum = trace = 7, product = det A = 10
Determinant (product of eigenvalues): 10 · Characteristic equation: λ² − (7)λ + (10) = 0
Shortcuts for eigenvalue questions
For triangular and diagonal matrices the eigenvalues are the diagonal entries. Symmetric matrices have real eigenvalues, skew-symmetric matrices have zero or purely imaginary ones, and orthogonal matrices have eigenvalues of modulus 1. The eigenvalues of A⁻¹ are 1/λ, of Aᵏ are λᵏ and of A + kI are λ + k. Checking the sum against the trace and the product against the determinant catches most arithmetic slips.
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Frequently asked questions
How do I check eigenvalues quickly in GATE?
Their sum equals the trace (sum of the diagonal) and their product equals the determinant. For a triangular matrix the eigenvalues are the diagonal entries.
Can a real matrix have complex eigenvalues?
Yes, they come in conjugate pairs a ± bi, as for the rotation matrix [[0, −1], [1, 0]] with eigenvalues ±i. A real symmetric matrix always has real eigenvalues.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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Scientific Converters. (2026, October 15). Eigenvalue Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/eigenvalue-calculator/
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Scientific Converters, "Eigenvalue Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/eigenvalue-calculator/