Characteristic polynomial: p(λ) = det(A − λI). For 2×2: λ² − tr(A)λ + det(A). Roots = eigenvalues. Coefficients encode: sum of eigenvalues = trace, product = determinant. Cayley-Hamilton: every matrix satisfies its own characteristic polynomial: p(A) = 0.
How to Use
Enter a matrix:
- A: Square matrix
- Output: p(λ) coefficients
- Roots: Eigenvalues
Cayley-Hamilton
Every matrix satisfies its characteristic polynomial: if p(λ) = λ² − tr(A)λ + det(A), then A² − tr(A)·A + det(A)·I = 0. This allows computing A⁻¹ = (tr(A)·I − A)/det(A) for 2×2.
Coefficient Meanings
- Leading: (−1)ⁿ
- λⁿ⁻¹ coefficient: (−1)ⁿ⁻¹·trace
- Constant: det(A)
- Sum of k×k minor determinants
Step-by-Step Instructions
- 1Enter a matrix.
- 2View polynomial.
- 3Find eigenvalues.
- 4Check trace = sum(λᵢ).
- 5Check det = prod(λᵢ).