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The minimal polynomial divides the characteristic polynomial,
Statement
For every endomorphism of a finite-dimensional vector space,
Facts & Assumptions
Given: A finite-dimensional endomorphism with minimal polynomial and characteristic polynomial (The basis-independent characteristic polynomial of an endomorphism of a finite-dimensional space, including in dimension zero).
Cayley–Hamilton states (Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, ).
A polynomial annihilates if and only if it is divisible by (The annihilator ideal is nonzero and has a unique monic generator; if and only if ).
Proof
By [L1], annihilates .
Therefore [L2] gives . In dimension zero both polynomials are , and the same argument applies.
Depends on
- The annihilator ideal is nonzero and has a unique monic generator; $p(T)=0$ if and only if $\mu_T\mid p$
- Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, $\chi_T(T)=0$
- The basis-independent characteristic polynomial $\chi_T$ of an endomorphism of a finite-dimensional space, including $\chi_T=1$ in dimension zero
Used by
- A characteristic polynomial that splits into distinct linear factors forces diagonalisability Corollary
- Every finite cyclic extension has a normal basis Theorem
- Over every extension field, a scalar is an eigenvalue of the extended matrix exactly when it is a root of the minimal polynomial Theorem
- The minimal and characteristic polynomials have exactly the same monic irreducible factors Theorem
Dependency tree · two levels
17 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, Corollary 4.10 (standard reference, not scraped)
- Sheldon Axler, Linear Algebra Done Right, 4th ed., Theorem 8.30 (standard reference, not scraped)