Universal Identities
نویسنده
چکیده
or equivalently (1.3) (ad− bc)(a′d′ − b′c′) = (aa′ + bc′)(cb′ + dd′)− (ab′ + bd′)(ca′ + dc′). A particular case of this is a = A, b = B, . . . , d′ = D′ in Z[A,B,C,D,A′, B′, C ′, D′]. In Section 2 we will describe how algebraic identities that make sense over all commutative rings can be proved by working only over C. This is a really significant idea! In Section 3 we’ll state two identities about determinants that will be proved by reduction to the complex case, including the Cayley-Hamilton theorem. Proofs will be given in Section 4, while Section 5 discusses some interesting consequences of the Cayley-Hamilton theorem. We will be dealing with multivariable polynomials, and will use an abbreviated notation for them. Rather than writing
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