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The annihilator ideal is nonzero and has a unique monic generator; if and only if
Statement
For every endomorphism of a finite-dimensional -vector space, is a nonzero ideal of and has a unique monic generator . For every ,
For the zero space, .
Facts & Assumptions
Given: An endomorphism of a finite-dimensional vector space over a field , and the annihilator set of The annihilator set ; once existence is proved, its unique monic generator is the minimal polynomial.
For every field , every ideal of is principal (For every field , is a principal ideal domain).
Cayley–Hamilton states for every finite-dimensional endomorphism (Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, ).
For the polynomial is monic of degree , and when ( is monic of degree ; for its coefficient is and its constant coefficient is , while ); and is by definition for any ordered basis , independently of the choice (The basis-independent characteristic polynomial of an endomorphism of a finite-dimensional space, including in dimension zero).
Every nonzero polynomial has a leading coefficient and is monic exactly when that coefficient is (Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree).
Proof
The zero polynomial lies in . If , then ; and if and , distributivity of the finite polynomial sums and associativity of composition give . Hence is an ideal.
By [L2], . Fix an ordered basis of the -dimensional ; by [L3], with , so is monic of degree and in particular nonzero. Thus the ideal is nonzero, including when , where .
If and are monic generators of this nonzero ideal, then and for some polynomials . Degrees force to be nonzero constants, and monicity forces both constants to be ; hence .
By [L1], write with . Multiplying by the inverse of its leading coefficient gives a monic generator by [L4], and step 1.3 shows that this monic generator is unique.
Finally, means , which is equivalent to for some , that is, . When , step 1.2 gives , so its monic generator is .
Depends on
- The annihilator set $\operatorname{Ann}(T)=\{p\in F[x]:p(T)=0\}$; once existence is proved, its unique monic generator $\mu_T$ is the minimal polynomial
- For every field $F$, $F[x]$ is a principal ideal domain
- Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, $\chi_T(T)=0$
- $\chi_A(x)$ is monic of degree $n$; for $n\geq1$ its $x^{n-1}$ coefficient is $-\operatorname{tr}(A)$ and its constant coefficient is $(-1)^n\det(A)$, while $\chi_{0\times0}=1$
- The basis-independent characteristic polynomial $\chi_T$ of an endomorphism of a finite-dimensional space, including $\chi_T=1$ in dimension zero
- Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree
Used by
- Every idempotent endomorphism is diagonalisable and is projection onto its image along its kernel Corollary
- For split operators, Jordan blocks read off eigenspace multiplicities and both canonical polynomials Corollary
- The algebra F[T] generated by an endomorphism is isomorphic to F[x]/(μ_T) Corollary
- The minimal polynomial divides the characteristic polynomial, μ_T∣χ_T Corollary
- Semisimple endomorphisms as endomorphisms diagonalisable over an algebraic closure, and nilpotent endomorphisms Definition
- A nilpotent shift has minimal polynomial xⁿ and, for n>0, a single primary component Example
- Computing the minimal polynomial of an explicit 3×3 idempotent from its annihilating polynomials Example
- Minimal polynomials of scalar and diagonal endomorphisms, including the zero-dimensional case Example
- The identity and a nontrivial Jordan block have the same characteristic polynomial but different minimal polynomials Example
- For a matrix over a field, extending the scalar field does not change its minimal polynomial Lemma
- Some vector has vector annihilator equal to the minimal polynomial Lemma
- For finite-dimensional V, V_T is finitely generated and torsion, with annihilator generated by the minimal polynomial Proposition
- Polynomial evaluation commutes with restriction and invariant quotients Proposition
- The minimal polynomial of a restriction to an invariant subspace divides the original minimal polynomial Proposition
- The vector annihilator is the unique monic generator of Ann_T(v) and divides the minimal polynomial Proposition
- A cyclic vector exists exactly when the minimal and characteristic polynomials agree Theorem
- An endomorphism is diagonalisable if and only if its minimal polynomial is a product of distinct linear factors Theorem
- Characterisations of a nilpotent endomorphism Theorem
- Every finite cyclic extension has a normal basis Theorem
- Primary decomposition: the irreducible-power factors of μ_T split V into their invariant kernels Theorem
- The minimal polynomial is unchanged by choosing a matrix representation or replacing a matrix by a similar one Theorem
Cited to discharge well-definedness by The annihilator set Ann(T)={p∈ F[x]:p(T)=0}; once existence is proved, its unique monic generator μ_T is the minimal polynomial.
Dependency tree · two levels
23 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
- Keith Conrad, The Minimal Polynomial and Some Applications, §4, Theorem 4.4 (standard reference, not scraped)
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §5B (standard reference, not scraped)