📐 Characteristic Polynomial Calculator
Calculate the characteristic polynomial det(A - λI) of a matrix
How to Use This Calculator
Select Matrix Size
Choose 2×2 or 3×3 matrix size.
Enter Matrix Elements
Input all elements of your square matrix.
Calculate
Click to compute det(A - λI), the characteristic polynomial.
Find Eigenvalues
The roots of the characteristic polynomial are the eigenvalues of the matrix.
Formula
det(A - λI) = 0
Where I is the identity matrix and λ represents eigenvalues
For 2×2 Matrix:
If A = [a b; c d], then:
det(A - λI) = det([a-λ b; c d-λ]) = (a-λ)(d-λ) - bc
= λ² - (a+d)λ + (ad - bc)
= λ² - trace(A)λ + det(A)
For 3×3 Matrix:
det(A - λI) = -λ³ + trace(A)λ² - (sum of 2×2 minors)λ + det(A)
General n×n:
p(λ) = (-1)ⁿ(λⁿ - c₁λⁿ⁻¹ + c₂λⁿ⁻² - ... ± cₙ)
where cᵢ are coefficients related to traces and determinants of minors
About Characteristic Polynomial Calculator
The Characteristic Polynomial Calculator finds the characteristic polynomial p(λ) = det(A - λI) of a square matrix A. The roots of this polynomial are the eigenvalues of the matrix, making it fundamental in linear algebra and matrix analysis.
When to Use This Calculator
- Eigenvalue Problems: Find eigenvalues by solving p(λ) = 0
- Linear Algebra: Study matrix properties and diagonalization
- Differential Equations: Solve systems of linear differential equations
- Physics: Quantum mechanics, vibration analysis, stability
- Engineering: Control theory, signal processing, structural analysis
Why Use Our Calculator?
- ✅ Direct Calculation: Computes det(A - λI) automatically
- ✅ Polynomial Form: Shows characteristic polynomial in standard form
- ✅ Coefficients: Displays all polynomial coefficients
- ✅ Educational: Helps understand eigenvalue theory
- ✅ Accurate: Precise symbolic and numeric computation
- ✅ Free: No registration required
Key Concepts
- Characteristic Polynomial: p(λ) = det(A - λI) is a polynomial in λ
- Eigenvalues: Roots of p(λ) = 0 are the eigenvalues
- Cayley-Hamilton Theorem: Every matrix satisfies its own characteristic polynomial: p(A) = 0
- Trace and Determinant: For 2×2, p(λ) = λ² - trace(A)λ + det(A)
- Degree: For n×n matrix, polynomial has degree n
Applications
Finding Eigenvalues: Solve p(λ) = 0 to find all eigenvalues λᵢ of matrix A.
Stability Analysis: Eigenvalues determine stability in differential equations and control systems.
Diagonalization: If eigenvalues are distinct, matrix can be diagonalized.
Frequently Asked Questions
What is a characteristic polynomial?
The characteristic polynomial p(λ) = det(A - λI) is a polynomial whose roots are the eigenvalues of matrix A. It's fundamental to finding eigenvalues.
How do I find eigenvalues from the characteristic polynomial?
Set p(λ) = 0 and solve for λ. The solutions are the eigenvalues. For example, if p(λ) = λ² - 5λ + 6, then (λ-2)(λ-3) = 0, so eigenvalues are 2 and 3.
What's the relationship to trace and determinant?
For a 2×2 matrix, p(λ) = λ² - trace(A)λ + det(A). The constant term is det(A), and the λ coefficient is -trace(A).
What is Cayley-Hamilton theorem?
Every square matrix satisfies its own characteristic polynomial: if p(λ) = det(A - λI), then p(A) = 0 (zero matrix).
Can I use this for any matrix size?
This calculator supports 2×2 and 3×3 matrices. For larger matrices, the characteristic polynomial becomes more complex but follows the same principle.
Why is it written as det(A - λI)?
Subtracting λI shifts all diagonal elements by -λ. The determinant of this modified matrix, set to zero, gives the eigenvalue equation: (A - λI)v = 0 has non-trivial solutions only when det(A - λI) = 0.