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Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial,
Statement
For every endomorphism of a finite-dimensional vector space,
Facts & Assumptions
Given: A finite-dimensional -vector space and .
The operator characteristic polynomial is computed from any representing matrix and equals in dimension zero (The basis-independent characteristic polynomial of an endomorphism of a finite-dimensional space, including in dimension zero); polynomial evaluation is (Polynomial evaluation at an endomorphism: ).
The adjugate consists of signed minors (Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring), and over a commutative ring (For every positive-sized square matrix over a commutative ring, ).
Equality of polynomials is equality of all coefficients (Finitely supported coefficient sequences and trimmed finite coefficient lists define the same formal polynomials).
Matrix representation is injective and sends sums and scalar multiples to matrix operations ( is a vector-space isomorphism ) and composition to matrix multiplication ().
Proof
If , [L1] gives , while is the unique endomorphism of the zero space; hence .
Suppose , choose a basis, and put . Each entry of has degree at most by [L2], so write ; also write , where .
Expanding the adjugate identity [L2] and comparing coefficients using [L3] gives , for , and .
Multiply the equation indexed by on the left by , include the first equation at and the last identity at , and add. The terms telescope against the terms of the preceding recurrence, leaving .
By [L1] and [L4], the left side of step 3.1 is the representing matrix of ; injectivity of matrix representation therefore gives .
Steps 1.1 and 4.1 prove Cayley-Hamilton in every finite dimension without treating substitution into a matrix-coefficient polynomial as a ring homomorphism.
Depends on
- The basis-independent characteristic polynomial $\chi_T$ of an endomorphism of a finite-dimensional space, including $\chi_T=1$ in dimension zero
- Polynomial evaluation at an endomorphism: $p(T)=\sum_k a_kT^k$
- Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring
- For every positive-sized square matrix over a commutative ring, $A\operatorname{adj}(A)=\operatorname{adj}(A)A=\det(A)I$
- Finitely supported coefficient sequences and trimmed finite coefficient lists define the same formal polynomials
- $T\mapsto[T]_{\mathcal B}^{\mathcal C}$ is a vector-space isomorphism $\mathcal L(V,W)\cong M_{m\times n}(F)$
- $[S\circ T]_{\mathcal B}^{\mathcal D}=[S]_{\mathcal C}^{\mathcal D}[T]_{\mathcal B}^{\mathcal C}$
Used by
- For an invertible matrix over a field, Cayley-Hamilton makes every matrix-power entry and trace sequence linearly recurrent 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
- Cayley-Hamilton reduces every power of beginpmatrix1&11&0 endpmatrix to a linear combination of A and I Example
- FALSE: substituting a fixed matrix A for x defines a ring homomorphism Mₙ(F[x])→ Mₙ(F) False statement
- A cyclic vector exists exactly when the minimal and characteristic polynomials agree Theorem
- Characterisations of a nilpotent endomorphism Theorem
- global regular functions projective variety Theorem
- The annihilator ideal is nonzero and has a unique monic generator; p(T)=0 if and only if μ_T∣ p Theorem
Dependency tree · two levels
28 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
- M. Khovanov, Linear Algebra II notes, §6 (standard reference, not scraped)
- The Stacks Project, Lemma 10.16.1 (05G6) (standard reference, not scraped)