Quasi-Rational Canonical Forms of a Matrix over a Number Field

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Avace Lea Algeba & Matx Theoy, 08, 8, -0 http://www.cp.

1 Avace Lea Algeba & Matx Theoy, 08, 8, -0 ISSN Ole: ISSN Pt: X Qua-Ratoal Caocal om of a Matx ove a Numbe el Zhueg Wag *, Qg Wag, Na Q School of Mathematc a Stattc, Yacheg Teache Uvety, Yacheg, Cha How to cte th pape: Wag, Z.D., Wag, Q. a Q, N. (08) Qua-Ratoal Caocal om of a Matx ove a Numbe el. Avace Lea Algeba & Matx Theoy, 8, Receve: Octobe 8, 07 Accepte: Jauay 7, 08 Publhe: Jauay 0, 08 Abtact A matx mla to Joa caocal fom ove the complex fel a the atoal caocal fom ove a umbe fel, epectvely. I th pape, we futhe tuy the atoal caocal fom of a matx ove ay umbe fel. We ftly cu the elemetay vo of a matx ove a umbe fel. The, we gve the qua-atoal caocal fom of a matx by combg Joa a the atoal caocal fom. ally, we how that a matx mla to t qua-atoal caocal fom ove a umbe fel. Copyght 08 by autho a Scetfc Reeach Publhg Ic. Th wo lcee ue the Ceatve Commo Attbuto Iteatoal Lcee (CC BY 4.0). Ope Acce Keywo Matx, Joa Caocal om, Ratoal Caocal om, Qua-Ratoal Caocal om. Itoucto A matx mla to Joa caocal fom ove the complex fel a the atoal caocal fom ove a umbe fel, epectvely. Thu, Joa a the atoal caocal fom of a matx ove the complex fel ae mla. Recetly, Rajabalpou [] tue the ymmetzato of the Joa caocal fom, Abo et al. [] a Baoe et al. [3] cue the elato betwee the egetuctue a Joa caocal fom. Moeove, L [4] cue the popety of the atoal caocal fom of a matx, Lu [5] gave out a cotuctve poof of extece theoem fo atoal fom, a Rajabalpou [6] vetgate the atoal caocal fom va the plttg fel. I th pape, we futhe tuy the atoal caocal fom ove ay umbe fel. We ftly cu the cocept of elemetay vo of a matx ove ay umbe fel. The, we gve the qua-atoal caocal fom of a matx by combg Joa a the atoal caocal fom. ally, we how that a matx mla to t qua-atoal caocal fom ove ay umbe fel. DOI: 0.436/alamt Ja. 0, 08 Avace Lea Algeba & Matx Theoy

2 Z. D. Wag et al.. Joa a Ratoal Caocal om Gve a matx A, t a teetg wo to f a mple matx that mla to A. We ow that uch a mple matx Joa caocal fom o the atoal caocal fom of A. Lemma.. ([7], pp 44-47) A matx A mla ove the complex fel to Joa caocal fom of t, uch Joa caocal fom uque up to a eaagemet of the oe of t chaactetc value,.e., A mla to the qua-agoal matx of oe whee J J J = J c c 0 0 J = c m m calle a elemetay Joa matx wth chaactetc value m + m + + m =. a Defto.. o a o-cala moc polyomal ove a umbe fel P, the matx a 0 0 a B = 0 0 a 0 0 a c, =,,, λ = λ + aλ + + a calle the compao matx o obeu matx of the moc polyomal. A polyomal matx, o λ-matx, a ectagula matx A λ whoe elemet ae polyomal λ ( 0 j j j j ) A λ = a λ = a λ + a λ + + a. Hee the laget of the egee of the polyomal aj. Two polyomal matce A a B λ ae calle equvalet f oe of them ca be obtae fom the othe by mea of ome elemetay opeato. A abtay ectagula polyomal matx equvalet to a caocal matx a a a DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy

3 Z. D. Wag et al. whee the polyomal a a a,,, ae ot etcally equal to zeo a each vble by the peceg oe. Let A be a polyomal matx of a,.e., the matx ha mo of oe ot etcally equal to zeo, but all the mo of oe geate tha ae etcally equal to zeo λ. We eote by D j the geatet commo vo of all the mo of oe j A ( j =,,, ). It eay to ee that the ee D, D,, D each polyomal vble by the peceg oe (ee [8], pp 39-40). A eay vefcato how mmeately that the elemetay opeato chage ethe the a of A o the polyomal D, D,, D. Thu, =, the coepog quotet wll be eote by D D λ = λ, λ =,, λ = D D vaat ue elemetay opeato a D = = = a, a,, a. The polyomal,,, of the λ-matx A. ae calle the vaat polyomal Defto.. ([8], pp 44-45) Let A= ( a j ) be a matx. We fom t chaactetc matx λ a a a a λ a a λe A =. a a λ a The chaactetc matx a λ -matx of a. It vaat polyomal D λ D λ λ λ =, =,, = D D D ae calle the vaat polyomal of the matx A. It eay to ee that the vaat polyomal of the compao matx B of the moc polyomal ae,,,. Defto.3. The followg qua-agoal matx B B C = B calle the ect um of the compao matce B of o-cala moc polyomal,,, uch that + ve fo =,, a a to be atoal caocal fom. The vaat polyomal of the atoal caocal fom matx B Defto.3 ae DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy

4 Z. D. Wag et al.,,,,,,. Lemma.. ([7], pp 38-39, Theoem 5) A matx A mla ove a umbe fel P to oe a oly oe matx whch atoal caocal fom,.e., A mla to the qua-agoal matx B B C = B whee B, B,, B ae the compao matce of the o-cala vaat polyomal,,, of matx A. 3. The Elemetay Dvo of a Matx ove a Numbe el Let P be a umbe fel. The a o-cala moc polyomal P[ x ] ca be factoe a a pouct of moc eucble polyomal P[ x ] oe a, except fo oe, oly oe way. I the factozato of a gve o-cala moc polyomal f ( x ), ome of the moc eucble facto may be epeate. If p ( x) p ( x) p ( x),,, ae the tct moc eucble polyomal occug th factozato of f x, the = f x p x p x p x beg the umbe of tme the eucble polyomal the expoet p x occu the factozato. Th ecompoto alo clealy uque, a calle the pmay ecompoto of f ( x ). Theoem 3.. If B the compao matx of the moc polyomal = λ + λ + + whee p a a B E B C = = E B E = ( cj ).e., c = c = = c = +, +,, + the the vaat polyomal of C ae,,, p. DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy

5 Z. D. Wag et al. Poof. Let B E B C E C c E B ( j ) = λ = = The c = c3 = = c, = c c c, c c c a a mo of oe 3 3 3, c c c,,, = ±.. Thu, D λ = = D λ = D λ = C., By Laplace Theoem, we have that λ λ D = E C = E B = p. Theefoe, the vaat polyomal of C ae The theoem pove.,,, p. λ a the chaactetc The matx C calle the atoal bloc of p polyomal of C pecely the lat vaat polyomal p of C. We ecompoe the vaat polyomal,,, of the λ-matx A to eucble facto ove the umbe fel P = p p p, = p p p, = p p p. Hee, p, p,, p ove P (a wth hghet coeffcet ) that occu a All the powe amog ae all the tct eucble polyomal,,, j j j, j =,,.,,, p p p a fa a they ae tct fom, ae calle the elemetay vo of the λ-matx A the umbe fel P. Theoem 3.. Aume that = S a polyomal of egee, whee p p p S DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy

6 Z. D. Wag et al. p, p,, p ae the tct moc eucble polyomal a,,, ae all potve tege. If C, C,, C ae, epectvely, the atoal bloc of S,,, p p p S the the vaat polyomal of the qua-agoal matx ae,,,. Poof. It eay to ee that C C = C C C = C a the vaat polyomal of C ( =,,, ),,, p by elemetay opeato of λ-matce, C ca be tafome to the caocal fom (ee [8], pp 40-4) C a be futhe tafome to. Thu, = p p =. p o λ-matx, we have that D 3 p λ p λ p λ p λ p λ p p ( 3 ) =,,, = Thu, = = D p p p. = = = = D D D, a the vaat polyomal of The theoem pove. D λ D λ D λ. D λ D λ D λ, =,, =, = DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy

7 Z. D. Wag et al. 4. Qua-Ratoal om of a Matx By combg Joa caocal fom ove the complex fel a the atoal caocal fom ove a umbe fel a ug the atoal bloc of p, we gve the qua-atoal caocal fom of a matx ove a umbe fel. Theoem 4.. If the vaat polyomal of a matx A ove a umbe fel P ae,,,,,,,,, ae the coepog matce of the o-cala vaat polyomal,, Theoem 3., the matx A mla ove the umbe fel P to the qua-agoal matx Poof. It eay to vefy that G =. G = a the vaat polyomal of ( =,,, ) ae,,,. Thu, by elemetay opeato of λ-matce, ca be tafome to the caocal fom a G be futhe tafome to G = =. By techagg ow o colum of G, G ca be tafome to. DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy

8 Z. D. Wag et al. Thu, G a A have ame caocal fom,.e., G a A ae equvalet. Theefoe, A a G ae mla. The theoem pove. The qua-agoal matx G Theoem 4. calle the qua-atoal caocal fom of matx A. Notg that ( =,,, ) Theoem 4. the ect um of the atoal bloc of ome elemetay vo of matx A, we ee that thee lttle bloc matce appeag the qua-atoal caocal fom of A ae pecely the atoal bloc of all elemetay vo of A. Thu, f we f all elemetay vo m,,, p p p m of A a the coepog atoal bloc,,, m, the λe m equvalet to p =. m pm By techagg ow o colum, we ow that equvalet to G G equvalet to Thu, =. DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy

9 Z. D. Wag et al. equvalet to Theefoe, A a the qua-agoal matx ae mla. m Smla to Joa caocal fom of a matx ove the complex fel, f we f all elemetay vo of a matx ove a umbe fel a atoal bloc of thee elemetay vo, the the ect um of thee atoal bloc pecely the qua-atoal caocal fom of the matx. Of coue, the qua-atoal caocal fom of a matx ot uque. But, the qua-atoal caocal fom uque up to a eaagemet of the oe of atoal bloc. 5. Cocluo I th pape, we futhe tuy the atoal caocal fom ove ay umbe fel a gve the qua-atoal caocal fom of a matx by combg Joa a the atoal caocal fom. Ule the compao matce the atoal caocal fom of a matx A [4] [5] [7], thee lttle bloc matce the qua-atoal caocal fom of a matx A ae the atoal bloc of elemetay vo of A a ot the compao matce of the o-cala vaat polyomal of A. Acowlegemet Th wo fue by the laghp Majo Developmet of Jagu Hghe Eucato Ittuto (PPZY05C) a College Stuet Pactce Iovato Tag Pogam ( Y). Refeece [] Rajabalpou, M. (07) A Symmetzato of the Joa Caocal om. Lea. DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy

10 Z. D. Wag et al. Algeba a It Applcato, 58, [] Abo, H., Elu, D., Kahle, T. a Peteo. C. (06) Egecheme a the Joa Caocal om. Lea Algeba a It Applcato, 496, [3] Baoe, M., Lma, J.B. a Campello e Souza, R.M. (06) The Egetuctue a Joa om of the oue Tafom ove el of Chaactetc a a Geealze Vaemoe-Type omula. Lea Algeba a It Applcato, 494, [4] L, S.G. (05) The Stuy o the Popety of Matx Ratoal Caocal om a It Applcato. College Mathematc, 3, (I Chee) [5] Lu, X.Z. (006) A Cotuctve Poof of Extece Theoem fo Ratoal om. College Mathematc,, 5-8. (I Chee) [6] Rajabalpou, M. (03) The Ratoal Caocal om va the Splttg el. Lea Algeba a It Applcato, 439, [7] Hoffma, K. a Kuze, R. (97) Lea Algeba. Eto, Petce-Hall, Ic., Eglewoo Clff. [8] Gatmache,.R. (960) The Theoy of Matce. AMS Chelea Publhg, New Yo. DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy

such that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1

such that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1 Scholas Joual of Egeeg ad Techology (SJET) Sch. J. Eg. Tech. 0; (C):669-67 Scholas Academc ad Scetfc Publshe (A Iteatoal Publshe fo Academc ad Scetfc Resouces) www.saspublshe.com ISSN -X (Ole) ISSN 7-9