The basis-independent characteristic polynomial ${\chi }_T$ of an endomorphism of a finite-dimensional space, including ${\chi }_T=1$ in dimension zero

Depends on

• For $A\in M_n(F)$, the characteristic polynomial is $\chi_A(x)=\det(xI_n-A)$ when $n\geq1$, with $\chi_A(x)=1$ for the unique $0\times0$ matrix
• Similar matrices have the same characteristic polynomial
• Coordinate columns $[v]_{\mathcal B}$ and matrices $[T]_{\mathcal B}^{\mathcal C}$ of linear maps relative to ordered bases
• Similarity is an equivalence relation, and two matrices represent the same endomorphism in two bases exactly when they are similar

Used by

• Every endomorphism of a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue Corollary
• For an invertible matrix over a field, Cayley-Hamilton makes every matrix-power entry and trace sequence linearly recurrent Corollary
• If χ_T splits over F, every eigenvalue of χ_T(T) is 0 Corollary
• Let K be a field, p≥1, and A∈ Mₚ(K). If χ_A(t)=∏_i<p(t-λᵢ) in K[t], then the transfer-matrix trace series is ∑_i<p(1-λᵢ x)⁻¹ Corollary
• The inverse of an invertible finite-dimensional endomorphism is a polynomial in that endomorphism Corollary
• The minimal polynomial divides the characteristic polynomial, μ_T∣χ_T Corollary
• The product of the invariant factors is the characteristic polynomial Corollary
• Algebraic multiplicity as the exponent of x-λ in χ_T, and geometric multiplicity as dim E_λ(T) Definition
• FALSE: A diagonalisable endomorphism must have a characteristic polynomial with distinct roots False statement
• For invariant W, χ_T=χ_T|_Wχ_T̄ Proposition
• An endomorphism is diagonalisable if and only if its characteristic polynomial splits and every eigenvalue's geometric multiplicity equals its algebraic multiplicity Theorem
• Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, χ_T(T)=0 Theorem
• Complexification preserves the characteristic and minimal polynomials of a finite-dimensional real operator Theorem
• Every finite cyclic extension has a normal basis Theorem
• For every finite-dimensional space, σ_F(T) is exactly the set of roots in F of χ_T Theorem
• If χ_T(x)=∏_i<n(x-λᵢ) in F[x], then det(T)=∏_i<nλᵢ: determinant is the product of the eigenvalues counted with algebraic multiplicity Theorem
• If χ_T(x)=∏_i<n(x-λᵢ) in F[x], then tr(T)=∑_i<nλᵢ: trace is the sum of the eigenvalues counted with algebraic multiplicity Theorem
• If χ_T(x)=∏_i<n(x-λᵢ) in F[x], then χ_p(T)(y)=∏_i<n(y-p(λᵢ)) for every p∈ F[x]: the eigenvalues of p(T) are p(λᵢ), counted with algebraic multiplicity Theorem
• The annihilator ideal is nonzero and has a unique monic generator; p(T)=0 if and only if μ_T∣ p Theorem
• The minimal and characteristic polynomials have exactly the same monic irreducible factors Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• H. Pinkham, Linear Algebra, §§12.1–12.4 (standard reference, not scraped)
• S. Axler, Linear Algebra Done Right, 4th ed., §9 (standard reference, not scraped)