Generalized Kronecker Product and Its Application

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1 Vol. 1, No. 1 ISSN: Generlize Kroneker Prout n Its Applition Xingxing Liu Shool of mthemtis n omputer Siene Ynn University Shnxi , Chin E-mil: lxx6407@163.om Astrt In this pper, we promote the efinition of Kroneker prout, n give its orresponing properties. As the pplition of generlize Kroneker prout, this pper shows the etermintion metho tht the lgeri opertion in finite set suits the ssoitive lw. Keywors: Generlize Kroneker prout, Assozitivitet, Deision onition 1. Introution Kroneker prout expresses speil prout of mtrix. The prout of mtrix A y m n n mtrix B y p q n e enote y A B, whih is mtrix y mp nq. Definition 1.(Kroneker prout) (BellmnR.,1970) The prout of mtrix A y m n n mtrix B y p q n e enote y A B, whih is efine s follows: 11 B 12 B 1n B 21 B 22 B 2n B A B [ i, j B] m1 B m2 B mn B Kroneker prout lso n e lle iret prout or tensor prout. Kroneker prout hs the following properties: 1). For A m n n B p q, generlly A B B A. 2). The Kroneker prout of ritrry mtrix n zero mtrix equls zero mtrix, i.e. A 0 0 A 0. 3). If α n β re onstnt, αa βb αβ(a B). 4). For A m n, B n k, C l p n D p q, AB CD (A C)(B D). 5). For A m n, B p q, C p q, A (B ± C) (A B) ± (A C), (B ± C) A (B A) ± (C A). Note: properties 1)-5) is referre from (Ro R,1971). 6).For A m n n B p q,(a B) T A T B T. 7). For A m n n B p q, rnk(a B) rnk(a)rnk(b). 8). For A m m n B n n, et(a B) (eta) n (etb) n. 9). For A m m n B n n, tr(a B) tr(a)tr(b). 10). For A m n, B m n, C p q n D p q,(a + B) (C + D) A C + A D + B C + B D. 92

2 Journl of Mthemtis Reserh Mrh, ). For A m n, B k l, C p q n D r s,(a B) (C D) A B C D. 12). For A m n, B k l, C p q,(a B) C A (B C). 13). For A m n, B p q, C n r n D q s,(a B)(C D) (A C)(B D). Note: properties 6)-13) is referre from (Brewer j w, 1978, ). But for the nee of the rel life n the mthemtis evelopment, the element of mtrix my not limit to numers, thus we introue new onept, lso nmely promoting the mtrix onept. 2. Definition of generlize Kroneker prout Definition 2. Suppose S e nonempty set, (S,, +) e n lger system, then A is mtrix in lgeri system (S,, +) if n only if A [ ij ] m n, ij S (i 1, 2,, m; j 1, 2,, n). Definition of Kroneker prout n e promote s the elow efinition. Definition 3(generlize Kroneker prout). The Kroneker prout of mtrix A y m n n mtrix B y p q in lger system (S,, +) n e enote y A B, whih is efine s follows: 11 B 12 B 1n B 21 B 22 B 2n B A B [ ij B] m1 B m2 B mn B Note: The generlize prout, ition n numer prout of mtrix re similr with the usul prout, ition n numer prout of mtrix. 3. Properties of generlize Kroneker prout Theorem 1. For A m n n B p q, generlly A B B A. Sine the numer operte is speil lgeri opertor, for restrite untenle proposition, it is lso untenle in generlize onition. Theorem 2. The Kroneker prout of ritrry mtrix n zero mtrix equls zero mtrix, i.e. A 0 0 A 0. The reson is s tht of theorem 1. Theorem 3. If (S, ) is ommuttive semi-group, n for ritrry α n β,(α A) (β B) (α β) (A B). This theorem is equivlent the operte of elements: (α i, j ) (β kl ) (α β) ( ij kl ), where ij [A] ij, kl [B] kl. Theorem 4. If (S, ) is ommuttive semi-group, n for A m n, B n k, C l p n D p q,(a B) (C D) (A C) (B D). This theorem is equivlent the operte of elements: ( ij kl ) ( tu wv ) ( i, j tu ) ( kl wv ), where ij [A] ij, kl [B] kl, tu [C] tu, wv [D] wv. Theorem 5. If (S, +, ) is ring, n for A m n, B p q, C p q, A (B ± C) (A B) ± (A C), (B ± C) A (B A) ± (C A). Theorem 6. If (S, ) is ommuttive lgeri system, n for A m n n B p q,(a B) T A T B T. Theorem 7. If (S, +, ) is ring n for A m n, B m n, C p q n D p q,(a+b) (C+D) A C+A D+B C+B D. This theorem is equivlent the operte of elements: ( ij + ij ) ( kl + kl ) ij kl + ij kl + ij kl + ij kl, where ij [A] ij, ij [B] ij, kl [C] kl, kl [D] kl. Theorem 8. If (S, ) is ommuttive semi-group, n for A m n, B k l, C p q n D r s,(a B) (C D) A B C D. This theorem is equivlent the operte of elements: ( ij kl ) ( tu wv ) ij kl tu wv, where ij [A] ij, kl [B] kl, tu [C] tu, wv [D] wv. Theorem 9. If (S, ) is semi-group, n for A m n, B k l, C p q,(a B) C A (B C). 93

3 Vol. 1, No. 1 ISSN: This theorem is equivlent the operte of elements: ( ij kl ) tu ij ( kl tu ), where ij [A] ij, kl [B] kl, tu [C] tu. Theorem 10. If (S, ) is ommuttive semi-group, n for A m n, B p q, C n r n D q s,(a B) (C D) (A C) (B D), where ij [A] ij, kl [B] kl, tu [C] tu, wv [D] wv. This theorem is equivlent the operte of elements: ( ij kl ) ( tu wv ) ( ij tu ) ( kl wv ). 4. Applition of generlize Kroneker prout [7],[8],[9],[10],[11] Theorem 12. Suppose S { 1, 2,, n } n (S, ) is n lgeri system, then the tle of lgeri opertor in S is s follows: <Figure1> Construt A [ 1 2 n ] T, then (S, ) is semi-group if n only if A (A A T ) (A A) A T. Theorem 13. Suppose S { 1, 2,, n } n (S, ) is n lgeri system, then the tle of lgeri opertor in S is s follows: <Figure1> Construt A [ 1 2 n ] T, then the lgeri opertor of S stisfies ssoitive lw if n if only A (A A T ) (A A) A T. Smple: Suppose set S {,,, }, the lgeri opertor in S is s follows: <Figure2> then whether is the lgeri opertor in S ssoitive? Proof: Esily, lgeri opertor in finite set S is lose, then (S, ) is lgeri system. Construt mtrix A, n the efinition of generlize Kroneker prout is s efinition 3, then [ ] A (A A T [ ] ) ( [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 94

4 Journl of Mthemtis Reserh Mrh, But (A A) A T ( ) [ ] [ ] [ ] [ ] 95

5 Vol. 1, No. 1 ISSN: Beuse [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]. [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]. A (A A T ) (A A) A T. From theorem 13, the lgeri opertor in S is suit for ssoitive lw. Referenes Bellmn R. (1970). Introution to Mtrix Anlysis. New York: MGrw-Hill. Ltkepohl H. (1996). Hnook of Mtries. New York:John Wiley n Sones. Ro C R,Mitr S K. (1971). Generlize Inverse of Mteies]. New York: John Wiley n Sones. Brewer J W. (1978). Kroneker prouts n mtrix lulus in system theory. IEEE Trns Ciruits n Systems, 25, Xin Zhng. (2004), Anlysis of mtrix n its pplition. Beijing: tsinghu university press, Anrews H C,Kne J. (1970). Kroneker mtries, omputetr implementtion, n generlize spetr. J Asso Comput Mh, 17: Alexner S T. (1986). Aptive Signl Proessing: Theory n Applitions. New York: Springer Verlg. Shoxue Liu. (1999). Bse of Moern Alger. Beijing: High Eution Press. Pinsn Wu. (1979). Moern Alger. Beijing: High Eution Press. Gunzhng Hu. (1999). Applition Moern Alger. Beijing: tsinghu university press. Pingtin Zhu, Bohong Li Et. (2001) Moern Alger. Beijing: Sientifi Press. 96

6 Journl of Mthemtis Reserh Mrh, 2009 Figure 1 Figure

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