For an n×n matrix A or linear operator, the characteristic polynomial is the monic polynomial p(λ)=det(λI−A) whose roots are the eigenvalues of A (over an algebraic closure), with multiplicities equal to algebraic multiplicities.

Principle

Degree equals matrix size n, coefficients are (up to sign) elementary symmetric polynomials in the eigenvalues: the coefficient of λ^{n−1} is −trace(A) and the constant term is (−1)^n det(A). The Cayley–Hamilton theorem asserts p(A)=0.

Demonstration

Example: For A = [[2,1],[0,3]] over R, p(λ)=det([[λ−2,−1],[0,λ−3]])=(λ−2)(λ−3)=λ^2−5λ+6; eigenvalues 2 and 3 are roots with algebraic multiplicity 1 each.

Misapplication

Confusing the characteristic polynomial with the minimal polynomial: the characteristic polynomial records algebraic multiplicities but may not capture the minimal annihilating polynomial or the sizes of Jordan blocks; assuming distinct roots from a degree count alone is unsafe over non-algebraically-closed fields.

Consequence

The characteristic polynomial encodes the spectrum and yields determinant and trace as coefficients; it is central to spectral analysis, to proving Cayley–Hamilton, and to invariant classification of linear operators up to similarity.

Reversal

Instead of using the characteristic polynomial to study spectrum, use the minimal polynomial or rational canonical form for finer structural information about cyclic decompositions and Jordan block sizes relevant to similarity classification.

Boundary

Defined for square linear operators; eigenvalues may lie in a field extension of the coefficient field, so roots must be considered over an algebraic closure; for infinite-dimensional operators the characteristic polynomial is not defined in general.

Semantic Tension

Tension between characteristic polynomial and minimal polynomial: both annihilate the operator in Cayley–Hamilton contexts but differ in degree and in the capture of Jordan structure; also tension between algebraic multiplicity of roots and geometric multiplicity (dimension of eigenspaces).

Synthesis

The characteristic polynomial is the monic determinant polynomial det(λI−A) of degree n that packages eigenvalues and relates directly to trace, determinant, and the Cayley–Hamilton relation, while leaving fine Jordan structure to the minimal polynomial.