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        AGeneralizedExtensionoftheCayly—HamiltonTheorem

        2016-05-09 09:24:52CUIChun-qiang
        校園英語(yǔ)·下旬 2016年4期

        CUIChun-qiang

        【Abstract】In this section, Our intention is to give a generalization of an extension of the Cayley-Hamilton theorem in the case of two n×n matrices in the formal matrix ring Mn(R; s).

        【Key words】formal matrix ring; the Cayley-Hamilton theorem; extension; 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 [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 n×n matrix over a commutative ring, then P(A)=0 where P (λ)= det(λIn-A) is the character-istic polynomial of A. The Cayley-Hamilton theorem 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 auxiliary de?nitions and lemmas.

        De?ne δijk=1+δik-δij-δjk where δik,δij,δjk are the Kronecker delta symbols. For s ∈ C(R), de?ne sijk=sδijk for all 1 ≤ i, j, k ≤ n. Note that

        De?nition 2.1. ([5], Def.2) Let R be a ring with s∈C(R) and let n ≥ 2. The set of all n×n matrices over R will be denoted by Mn(R; s), with usual 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 indeterminate over R and let R[x] be the poly-nomial ring. Then there exists a ring homomorphism Φx : Mn(R[x]; x)Mn(R[x]) given by (rij(x)) (x1-δij rij(x)).

        Given A =(aij) ∈ Mn(R; s), there is a matrix of Mn(R[x]; x), denoted by A,whose (i, j)-entry is aij. Then Φx(A) ∈ Mn(R[x]). Let adj(Φx(A)) be the adjoint matrix of Φx(A) in Mn(R[x]) and write adj(Φx(A)) = (fij(x)) . Since x is a non zero-divisor of R[x], one easily sees that there exists a unique gij(x) ∈ R[x] such that fij(x)= x1-δij gij(x). Denote adjx(A)=(gij(x)) ∈ Mn(R[x]; x) and call it the x-adjoint matrix of A. The matrix adjs(A) := (gij(s)) ∈ Mn(R; s) is called the 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; s) to be the characteristic polynomial of Φs(A) in Mn(R).

        Lemma 2.5. ([5], Th.34) (Cayley-Hamilton theorem) Let A ∈ Mn(R; s). If f(λ) ∈ R[λ] 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.

        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+ 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 equations:

        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 + 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:

        [1]F.R Chang.and C.N.Chan,The generalized Cayley-Hamilton theorem for stan-dard pencils; System and Control Lett.,vol.18(1992),179-182.

        [2]T.Kaczorek,An existence of the Cayley-Hamilton theorem for nonsquare block matrices and computation of the left and right inverses; Bull.Pol.Acad.Techn.Sci.,vol.43(1)(1995),49-56.

        [3]T.Kaczorek,New extensions of the Cayley-Hamilton theorem with applications; Proceeding of the 19th European Conference on Modelling and Simulation,2005.

        [4]I.Kaddoura and B.Mourad,A Not on a Generalization of an Extension of the Cayley-Hamilton Theorem,Intel.Math.Forum,3(17)(2008),829-836.

        [5]G.Tang,Y.Zhou,A class of formal matrix rings,Linear Algebra Appl.438(12)(2013),4672-4688.

        [6]L.F.Urrutia and N.Morales,The Cayley-Hamilton theorem for supermatrices,J.Phys.A:Math.Gen.,27(1994),1981-1997.

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