About

The characteristic polynomial of a square matrix A encodes its eigenvalues as roots. Miscalculating it propagates errors into stability analysis, control system design, and principal component extraction. This tool computes det(λI − A) using the Faddeev - LeVerrier algorithm, which recovers all n + 1 coefficients in O(n⁴) operations without symbolic expansion. Results are exact for integer entries and IEEE-754 accurate for decimal entries. The tool handles matrices from 1×1 up to 6×6.

Limitation: floating-point rounding may affect coefficients for ill-conditioned matrices with entries exceeding 10⁸. For matrices up to 3×3, eigenvalues are computed analytically. For 4×4 and above, eigenvalues are approximated numerically via QR iteration on the companion matrix. Pro tip: verify results by checking that the sum of eigenvalues equals tr(A) and their product equals det(A).

Formulas

The characteristic polynomial of an n × n matrix A is defined as:

p(λ) = det(λI − A) = λⁿ + c_n−1λⁿ⁻¹ + … + c₁λ + c₀

The Faddeev - LeVerrier algorithm computes the coefficients iteratively. Starting with M₀ = 0 (zero matrix), at each step k = 1, 2, …, n:

M_k = A ⋅ (M_k−1 + c_n−k+1 ⋅ I)

c_n−k = −1k tr(M_k)

Where p(λ) is the characteristic polynomial, λ is the eigenvalue parameter, I is the n × n identity matrix, A is the input square matrix, c_k are the polynomial coefficients, tr() denotes the matrix trace (sum of diagonal elements), det() denotes the determinant, and M_k are intermediate matrices in the recurrence. The leading coefficient of λⁿ is always 1 (monic polynomial).

Reference Data

Matrix Size	Polynomial Degree	Coefficients Count	Faddeev - LeVerrier Steps	Analytical Roots	Common Applications
1×1	1	2	1	Direct	Scalar eigenvalue
2×2	2	3	2	Quadratic formula	2D transformations, coupled oscillators
3×3	3	4	3	Cardano's method	3D rotations, stress tensors
4×4	4	5	4	Numerical (QR)	Spacetime metrics, control systems
5×5	5	6	5	Numerical (QR)	Quantum mechanics, graph Laplacians
6×6	6	7	6	Numerical (QR)	Finite element analysis, PCA
Key Relationships
c_n−1		−tr(A) (negative trace)
c₀		(−1)ⁿ det(A)
Cayley - Hamilton		p(A) = 0 (matrix satisfies its own polynomial)
Vieta's formulas		λ₁ + … + λ_n = tr(A)
Determinant		λ₁ ⋅ … ⋅ λ_n = det(A)
Similar matrices		Share the same characteristic polynomial
Symmetric real matrix		All eigenvalues are real
Orthogonal matrix		All eigenvalues have \|λ\| = 1
Nilpotent matrix		p(λ) = λⁿ
Identity matrix I_n		p(λ) = (λ − 1)ⁿ

Frequently Asked Questions

Cofactor expansion computes det(λI − A) symbolically, requiring O(n!) operations in the naive case. The Faddeev - LeVerrier algorithm avoids symbolic manipulation entirely by computing the polynomial coefficients numerically through matrix multiplications and traces. It runs in O(n⁴) time - n iterations each involving an O(n³) matrix product - making it vastly more efficient for n ≥ 4.

Repeated eigenvalues correspond to repeated roots of the characteristic polynomial. The algorithm computes coefficients correctly regardless of multiplicity. However, numerical root-finding for repeated roots is inherently ill-conditioned: a small perturbation in coefficients can shift root locations significantly. For a matrix with eigenvalue λ₀ of multiplicity m, expect reduced precision in the reported eigenvalues proportional to the condition number raised to 1/m.

Yes. A real non-symmetric matrix may have complex conjugate eigenvalue pairs. The characteristic polynomial coefficients remain real, but the roots may be complex. For 2×2 and 3×3 matrices, the analytic solvers detect negative discriminants and report complex eigenvalues in a + bi form. For larger matrices, the QR iteration on the companion matrix detects 2×2 blocks on the quasi-upper-triangular Schur form, extracting complex pairs from those blocks.

Setting λ = 0 in det(λI − A) yields det(−A) = (−1)ⁿ det(A). Since p(0) = c₀ by definition, the constant term directly encodes the determinant scaled by the sign factor. This provides a built-in consistency check: compute det(A) separately and verify it matches (−1)ⁿ · c₀.

The Faddeev - LeVerrier recurrence accumulates products and traces. For entries of magnitude M in an n×n matrix, intermediate values can reach Mⁿ. IEEE-754 double precision provides roughly 15-16 significant digits. For a 6×6 matrix with entries around 10⁸, intermediate values approach 10⁴⁸, nearing the overflow boundary of ~1.8 × 10³⁰⁸ but suffering significant precision loss. Scale the matrix before computation if entries span many orders of magnitude.

Yes. The Cayley - Hamilton theorem states that every square matrix satisfies its own characteristic polynomial: p(A) = 0 (zero matrix). You can verify the result by substituting the matrix A into the computed polynomial. Evaluate c_n · Aⁿ + c_{n−1} · A^{n−1} + … + c₀ · I. The result should be the zero matrix within floating-point tolerance, typically ‖p(A)‖ < 10⁻¹⁰ for well-conditioned integer matrices.