When the domain and codomain of a linear map coincide, we gain additional algebraic structure: the ability to compose maps with themselves. Recall from Chapter 5 that a linear map is a linear operator, or endomorphism, of , and that is a ring under addition and composition, with identity (Proposition 5.1.5). Composition is also compatible with scalars, ; one says that is a -algebra. For , choosing a basis gives a ring isomorphism (Corollary 5.2.10), and the ring is noncommutative once .
Since endomorphisms can be composed, we can form powers and evaluate polynomials.
Definition 7.2.1 (Powers and polynomials of operators). For and : For a polynomial , define For we often write for .
Proposition 7.2.2 (Evaluation homomorphism). The map defined by is a ring homomorphism. That is:
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In particular, for all : polynomials in the same operator commute.
Proof. (a) and (c) are immediate. For (b), let and . Then where . Since and composition is bilinear, The last sentence follows from (b) because in .∎
Note. A factorization of a polynomial therefore gives a factorization of the operator: if , then , with the factors in any order. In the language of Chapter 10, the rule makes a module over the ring . The polynomials with form the kernel of ; in Section 7.9 we show that they are exactly the multiples of a single monic polynomial, the minimal polynomial of .