What Is a Characteristic Polynomial?
The characteristic polynomial of a square matrix A (size n × n) is defined as p(λ) = det(A − λI), where I is the identity matrix and det denotes the determinant. This polynomial encodes crucial information about the matrix's eigenvalues, which are the roots of p(λ).
Different conventions exist: some authors define it as det(λI − A) instead. The two definitions differ by a factor of (−1)ⁿ, so they yield identical roots. For even-sized matrices, both formulations produce the same polynomial; for odd sizes, the coefficients flip sign.
The degree of the characteristic polynomial always equals the size of the matrix. For a 3×3 matrix, you'll get a cubic polynomial; for a 4×4 matrix, a quartic.
Characteristic Polynomial Formulas
For a 2×2 matrix, the calculation is straightforward. For larger matrices, the determinant computation becomes increasingly complex, involving multiple cofactor expansions or alternative methods like the Rule of Sarrus (for 3×3).
2×2 matrix:
p(λ) = λ² − tr(A)λ + det(A)
where tr(A) = a + d (trace) and det(A) = ad − bc
3×3 matrix:
p(λ) = −λ³ + tr(A)λ² − S₂λ + det(A)
where S₂ is the sum of all 2×2 principal minors
General form (n × n matrix):
p(λ) = (−1)ⁿλⁿ + (−1)ⁿ⁻¹S₁λⁿ⁻¹ + ... + Sₙ
where Sₖ is the sum of all k×k principal minors
λ— The polynomial variable (lambda), representing potential eigenvaluestr(A)— Trace of the matrix: sum of diagonal elementsdet(A)— Determinant of the matrixSₖ— Sum of all k×k principal minors of A
How to Find the Characteristic Polynomial Step by Step
For a 2×2 matrix: Write out (a − λ)(d − λ) − bc and expand to get λ² − (a + d)λ + (ad − bc).
For a 3×3 matrix: Form the matrix (A − λI), then compute its determinant using cofactor expansion or the Rule of Sarrus. This involves three 2×2 minor determinants. Collect like terms in powers of λ.
For larger matrices: The algebraic effort scales significantly. Use cofactor expansion along a row or column, or apply row operations to simplify. Computing principal minors becomes tedious by hand, which is why computational tools are invaluable.
Verification: Check that the constant term equals (−1)ⁿ times det(A), and the leading coefficient is (−1)ⁿ.
Key Properties and Applications
Invertibility: A matrix is invertible if and only if its characteristic polynomial has a non-zero constant term (equivalently, det(A) ≠ 0).
Eigenvalues: The roots of p(λ) are precisely the eigenvalues of A. Their multiplicities as roots are called algebraic multiplicities.
Symmetry: A matrix and its transpose share the same characteristic polynomial: det(A − λI) = det(Aᵀ − λI).
Similar matrices: If two matrices are similar (related by a change of basis), they have identical characteristic polynomials.
Cayley–Hamilton theorem: Every square matrix satisfies its own characteristic polynomial equation: p(A) = 0.
Common Pitfalls and Tips
Avoid these typical mistakes when computing or interpreting characteristic polynomials.
- Sign conventions matter — Different authors use det(A − λI) or det(λI − A). These differ by (−1)ⁿ. Always check which definition your course or reference uses, especially when comparing results with textbooks or software.
- Principal minors are not the same as cofactors — When computing the general polynomial formula, S₂ is the sum of all 2×2 principal minors (minors of the matrix itself), not the cofactors used in determinant expansion. This distinction is crucial for correct coefficients.
- Algebraic versus geometric multiplicity — A root λ of p(λ) has an algebraic multiplicity (its order as a root of the polynomial) and a geometric multiplicity (the dimension of the corresponding eigenspace). These can differ, and the distinction matters for diagonalizability.
- Complex eigenvalues occur in conjugate pairs — For real matrices, non-real eigenvalues always appear as complex conjugate pairs. This means the characteristic polynomial has real coefficients even if some roots are complex.