Eigenvalue Calculator (2×2)
Find the eigenvalues of a 2×2 matrix by solving the characteristic polynomial det(A - λI) = 0.
This free online eigenvalue calculator (2×2) provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Results
Trace (a + d)
7
Determinant (ad - bc)
10
Discriminant
9
Eigenvalue λ₁
5
Eigenvalue λ₂
2
How to Use This Calculator
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Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
When to Use This Calculator
- •Use the Eigenvalue Calculator (2×2) when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
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About Eigenvalue Calculator (2×2)
Eigenvalues are scalar values λ for which a matrix A has a nonzero vector v satisfying Av = λv. This means the matrix acts on the eigenvector v by simply scaling it by λ, without changing its direction. Eigenvalues reveal fundamental properties of a matrix: whether it stretches, compresses, or reflects space, and by how much. For a 2x2 matrix, the eigenvalues are the roots of a quadratic characteristic polynomial, making them easy to compute. Eigenvalues and eigenvectors are central to stability analysis in dynamical systems, principal component analysis in statistics, quantum mechanics, vibration analysis in engineering, and Google's PageRank algorithm.
The Math Behind It
Formula Reference
Characteristic Polynomial (2×2)
λ² - (a+d)λ + (ad-bc) = 0
Variables: λ = eigenvalue, trace = a+d, det = ad-bc
Eigenvalue Formula
λ = (trace ± √(trace² - 4·det)) / 2
Variables: Quadratic formula applied to the characteristic polynomial
Worked Examples
Example 1: Finding eigenvalues of a 2×2 matrix
Find eigenvalues of [[4, 1], [2, 3]]
Eigenvalues: λ₁ = 5, λ₂ = 2
Common Mistakes & Tips
- !Forgetting the negative sign in det(A - λI), leading to incorrect characteristic polynomial.
- !Miscalculating the discriminant (trace^2 - 4*det, not trace^2 - 4*trace*det).
- !Not checking results against trace and determinant relationships.
- !Assuming complex eigenvalues mean an error (they are valid for non-symmetric matrices).
Related Concepts
Used in These Calculators
Calculators that build on or apply the concepts from this page:
Frequently Asked Questions
Can eigenvalues be complex numbers?
Yes. When the discriminant is negative, the eigenvalues come in complex conjugate pairs a ± bi.
What is an eigenvector?
An eigenvector v corresponding to eigenvalue λ satisfies Av = λv. It is the direction that is only scaled (not rotated) by the matrix transformation.
What does a negative eigenvalue mean?
The corresponding eigenvector direction is reversed (reflected) by the transformation, in addition to being scaled by the absolute value.
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