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The adjugate of an operator on a positive-dimensional finite-dimensional vector space, defined by the adjugate matrix in any basis
Definition
Let be a linear operator on a finite-dimensional vector space over a field, with . Choose an ordered basis . The adjugate operator is the unique linear operator satisfying
where the matrix adjugate is defined in Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring. Existence and uniqueness follow because the matrix-representation map is an isomorphism ( is a vector-space isomorphism ).
This definition is independent of . Indeed, the two-sided change-of-basis formula gives (), and adjugate equivariance gives
(Over a commutative ring, for every invertible ), which is exactly the change-of-basis formula for the same operator.
Depends on
- Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring
- Over a commutative ring, $\operatorname{adj}(P^{-1}AP)=P^{-1}\operatorname{adj}(A)P$ for every invertible $P$
- $[T]_{\mathcal B'}^{\mathcal C'}=P_{\mathcal C'\leftarrow\mathcal C}[T]_{\mathcal B}^{\mathcal C}P_{\mathcal B\leftarrow\mathcal B'}$
- $T\mapsto[T]_{\mathcal B}^{\mathcal C}$ is a vector-space isomorphism $\mathcal L(V,W)\cong M_{m\times n}(F)$
Used by
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Sources
- Sheldon Axler, Linear Algebra Done Right, 4th ed. (standard reference, not scraped)