Characteristic Polynomial Calculator
Calculate the characteristic polynomial det(A - λI) of a matrix.
Matrix A
Characteristic Polynomial p(λ) = det(A - λI)
λ³ - 8λ² + 19λ - 12
Trace (tr A)
8.0000
Sum of eigenvalues
Determinant
12.0000
Product of eigenvalues
Coefficients
c3 = 1.000000 (coefficient of λ³)
c2 = -8.000000 (coefficient of λ²)
c1 = 19.000000 (coefficient of λ)
c0 = -12.000000 (coefficient of λ⁰)
Cayley-Hamilton Theorem
Every matrix satisfies its own characteristic equation: p(A) = 0
Characteristic Polynomial
Definition
p(λ) = det(A - λI)
Polynomial whose roots are the eigenvalues of A
Properties
- Degree equals matrix dimension
- Coefficient of λⁿ⁻¹ is -trace(A)
- Constant term is (-1)ⁿ det(A)