Characteristic Polynomial Calculator
Calculate the characteristic polynomial of square matrices with step-by-step solutions
Matrix Input
Characteristic Polynomial
Formula Used:
p(λ) = det(A - λI)
Result:
p(λ) = λ^2 - 5λ - 6
Step-by-step for 2×2:
For matrix A = [[2, 3], [4, 3]]
Trace tr(A) = 2 + 3 = 5
Determinant det(A) = (2)(3) - (3)(4) = -6
p(λ) = λ² - tr(A)λ + det(A)
Example Calculation
2×2 Matrix Example
Matrix A = [[2, 3], [4, 3]]
Formula: det(A - λI)
= det([[2-λ, 3], [4, 3-λ]])
= (2-λ)(3-λ) - 3×4
= λ² - 5λ - 6
Key Formulas
2×2 Matrix:
λ² - tr(A)λ + det(A)
3×3 Matrix:
-λ³ + tr(A)λ² - S₁λ + det(A)
where S₁ is sum of 2×2 minors
General:
Roots of the characteristic polynomial are the eigenvalues of the matrix
Understanding Characteristic Polynomials
What is a Characteristic Polynomial?
The characteristic polynomial of a square matrix A is defined as p(λ) = det(A - λI), where I is the identity matrix and λ is a variable. This polynomial plays a crucial role in linear algebra and matrix analysis.
Why is it Important?
- •The roots of the characteristic polynomial are the eigenvalues of the matrix
- •Used in solving differential equations and analyzing system stability
- •Essential for matrix diagonalization and similarity transformations
- •Applications in quantum mechanics, vibration analysis, and data science
Key Properties
Cayley-Hamilton Theorem
Every square matrix satisfies its own characteristic equation. If p(λ) is the characteristic polynomial of A, then p(A) = 0 (zero matrix).
Alternative Definitions
Some authors define the characteristic polynomial as det(λI - A). The two definitions differ by a factor of (-1)ⁿ, but have the same roots (eigenvalues).
Matrix Invariants
The coefficients relate to matrix invariants: the coefficient of λⁿ⁻¹ is ±tr(A), and the constant term is ±det(A).