1、AGeneralizedExtensionoftheCaylyHamiltonTheorem【Abstract】In this section, Our intention is to give a generalization of an extension of the Cayley-Hamilton theorem in the case of two nn matrices in the formal matrix ring Mn(R; s). 【Key words】formal matrix ring; the Cayley-Hamilton theorem; extension;
2、integral domain. 1. Introduction The Cayley-Hamilton theorem and its extensions have many applications in con-trol systems, electric circuit and many other areas (see 3, 6). The Cayley-Hamilton theorem has been extended to rectangular matrices, block matrices and to pairs of commuting matrices (see
3、1, 3, 2). The classical Cayley-Hamilton theorem states that every square matrix over a com-mutative ring satis?es its own characteristic equation. Precisely, if A is an nn matrix over a commutative ring, then P(A)=0 where P ()= det(In-A) is the character-istic polynomial of A. The Cayley-Hamilton th
4、eorem has been extended to matrices over the formal matrix ring Mn(R; s) in 5. The Jacobson radical, the center, the set of zero-divisors and the group of units of a ring R are denoted by J(R) , C(R) , Z(R) ,and U(R) respectively. 2. Extension of the Cayley-Hamilton theorem First, we begin with some
5、 auxiliary de?nitions and lemmas. De?ne ijk=1+ik-ij-jk where ik,ij,jk are the Kronecker delta symbols. For s C(R) , de?ne sijk=sijk for all 1 i, j, k n. Note that De?nition 2.1. (5, Def.2) Let R be a ring with sC(R) and let n 2. The set of all nn matrices over R will be denoted by Mn(R; s) , with us
6、ual addition of matrices and with multiplication de?ne below, for any A =(aij) , B =(bij) Mn(R; s) , We could prove that Mn(R; s) forms a ring, and Mn(R; s) is called a formal matrix ring over R de?ned by s. For instances, in M2(R; s) , and in M3(R; s). De?nition 2.2. (5, p.4685) Let x be an indeter
7、minate over R and let Rx be the poly-nomial ring. Then there exists a ring homomorphism x : Mn(Rx; x)Mn(Rx) given by (rij(x) ) (x1-ij rij(x) ). Given A =(aij) Mn(R; s) , there is a matrix of Mn(Rx; x) , denoted by A,whose (i, j)-entry is aij. Then x(A) Mn(Rx). Let adj(x(A) ) be the adjoint matrix of
8、 x(A) in Mn(Rx) and write adj(x(A) ) = (fij(x) ) . Since x is a non zero-divisor of Rx, one easily sees that there exists a unique gij(x) Rx such that fij( x)= x1-ij gij(x). Denote adjx(A)=(gij(x) ) Mn(Rx; x) and call it the x-adjoint matrix of A. The matrix adjs(A) := (gij(s) ) Mn(R; s) is called t
9、he s-adjoint matrix of A in Mn(R; s). Lemma 2.3. (5, Prop.32) Let A, B Mn(R; s).Then: (1)dets(A ? B)= dets(A)dets(B). (2)A ? adjs( A)= adjs(A) ? A = dets(A) ? In = dets(A)In. (3)A is invertible if and only if dets(A) U(R). De?nition 2.4. (5, Def.33) We de?ne the s-characteristic polynomial of A Mn(R
10、; s) to be the characteristic polynomial of s(A) in Mn(R). Lemma 2.5. (5, Th.34) (Cayley-Hamilton theorem) Let A Mn(R ; s). If f() R is the s-characteristic polynomial of A, then f(A)=0 in Mn(R; s). Theorem 2.6. Let A, B Mn(R; s) and let PA,B(x, y)= dets(xA - yB). If A ? B = B ? A, then PA,B(B, A)=0
11、. Proof: Note that PA,B(x, y) is a homogeneous polynomial of degree n in the two variables x and y. So we can assume that PA,B(x, y)= anxn+ an-1xyn-1 + ? + a1xyn-1+ a0yn. Let N be the s-adjoint matrix of the matrix xA - yB, that is N = adjs(xA - yB). Clearly, N can be written in the form N= Nn-1xn-1
12、+ Nn-2xn-2y + ? + N1xyn-2+N0yn-1, where Nn-1,Nn-2 ? ,N1,N0 Mn(R; s). Now, for any A Mn(R; s) , we have A ? adjs(A)= dets(A)In from lemma 2.3 (2). Thus (xA - yB) ? N = PA,B(x, y)In. Expanding the left-hand side and comparing the coe?cients of xiyj in both sides, we obtain the following n + 1 equation
13、s: Multiplying these equations to the left by Bn , Bn-1 ? A, Bn-2 ? A2 , ? , An respec-tively, and using the fact that A and B commute, we obtain the following equations: Hence, the terms on the left-hand side will cancel out leaving the zero matrix. The terms on the right-hand side add up to anBn +
14、 an-1Bn-1 ? A + an-2Bn-2 ? A2 + ? + a1B ? An-1 + a0An = PA,B(B, A)=0 The theorem is established. Remark 2.7. In the above theorem, if y =1 and A = In, then we obtain the classical Cayley-Hamilton theorem over formal matrix ring Mn(R; s). References: 1F.R Chang.and C.N.Chan,The generalized Cayley-Ham
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