PURE MATHEMATICS RESEARCH ARTICLE. Subjects: Science; Mathematics & Statistics; Advanced Mathematics; Algebra

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1 PURE MATHEMATICS RESEARCH ARTICLE Nturl rhotrix A.O. Isr * Rcivd: 27 July 206 Accptd: 05 Octobr 206 First Publishd: Octobr 206 *Corrspodig uthor: A.O. Isr, Dprtmt of Mthmtics, Ambros Alli Uivrsity, Ekpom 3000, Nigri E-mills: bdis@yhoo.co.uk, isro@ukpom.du.g Rviwig ditor: Lish Liu, Qufu Norml Uivrsity, Chi Additiol iformtio is vilbl t th d of th rticl Abstrct: This ppr tks look t st of rhotrix whos tris r ordrd turl umbrs. This rhotrix is clld th turl rhotrix. Proprtis of this st r xmid d th rsults r prstd. Sic th turl rhotrix R is ot ivrtibl, mid ivstigtio is md ito th cocpts of mior rhotrics of R, dtrmit fuctios (h(r)), codtrmit fuctio (codt(r)), d idx (ρ) of turl rhotrix. It ws foud tht codt(r) =ρh(r), d thir mthods of computtios r outlid for mthmticl richmt. Subjcts: Scic; Mthmtics & Sttistics; Advcd Mthmtics; Algbr Kywords: rhotrix; dtrmit fuctio; codtrmit fuctio d idx 200 Mthmtics subjct clssifictios: primry 5B99; scodry Itroductio A rhotrix A of dimsio thr is rhomboidl rry dfid s: A = whr, b, c, d, R. Th try c i A is clld th hrt of A, dotd s h(a). Th cocpt of rhotrix ws itroducd by Ajibd (2003) s xtsio of mtrix-trtios d mtrix-oitrts by Atssov d Sho (998). Th cocpt of rhotrics is still t its lmtry stg of dvlopmt. It is brly ovr dcd go. Sic its birth i 2003, my rsrchrs hv show itrst ABOUT THE AUTHORS A.O. Isr obtid first d scod dgrs i Mthmtics from th Uivrsity of Bi, Bi City, Nigri, i 999 d 2004, rspctivly. H ws mployd by Ambros Alli Uivrsity, Ekpom, Nigri i 2006, s lcturr d rsrchr i Mthmtics till dt. H got his PhD i Algbr from Fdrl Uivrsity of Agricultur, Abokut, Nigri, i 204. H spcilizs i Loop Thory, prcisly i Osbor loops. Btw 202 d 205, h hs publishd vrious rticls o fiit Osbor loops, Holomorphy of Osbor loops d Osbor loops of ordr 4. I 205, h pickd itrst i Rhotrix lgbr d is currtly workig o th clssicl d oclssicl rhotrics, d Rhotrix loops. But i this rticl grlizs th xmpls of turl rhotrix, itroducs th cocpt of dtrmit fuctio, codtrmit fuctio d idx of turl rhotrix. PUBLIC INTEREST STATEMENT A turl rhotrix is rhotrix whos lmts r ll coutbl umbrs. A rhotrix is systm of umbrs i rhomboid shp. Thy r similr to mtrics. Ths systms of umbrs form sts of rhotrics dpdig o th mout of umbrs i th systms. Ech st obys som mthmticl ruls dpdig o th typ of umbrs you hv i th st d bhv diffrtly. This work xmis th bhvior of rhotrix st whos lmts r coutbl umbrs. I rl lif, childr from diffrt fmilis bhv diffrtly. Thrfor, scitific study of ch fmily will improv itr hum rltioship d hlp to hrss th tlts ihrt i ch of th fmily. This is to illustrt th ssc of th scitific study of fmilis or sts of physicl proprtis roud us lik this work. Dtil itroductio, trsformtio ito mtrics d pplictios boud i litrtur. 206 Th Author(s). This op ccss rticl is distributd udr Crtiv Commos Attributio (CC-BY) 4.0 lics. Pg of 0

2 i dvlopig d xpdig this cocpt, most oft, i logy with th cocpts of mtrics usully through trsformtio tht covrts mtrix ito rhotrix d vic vrs (Ajibd, 2003; Si, 2008). O of such works ws th clssifictio of rhotrics ito sts d lgbric spcs by Mohmmd d Tll (202). Th ppr clssifis rhotrics ito turl rhotrix st, rl rhotrix st, complx rhotrix st, rtiol d irrtiol rhotrix sts. Thus, thir work hs ctully opd up diffrt brchs of studyig rhotrics. Thrfor, this rticl is pickig o th first brch, th turl rhotrix st. Furthrmor, i this ppr, most of ths proprtis of turl rhotrics will b xmid without cssrily hvig to go through trsformtio. Though, if d b, ths proprtis would lwys b i coformity to o trsformtio or th othr. This c b xmid. Dfiitio. Mohmmd d Tll (202) A rhotrix st is clld turl rhotrix st if its rhotric tris blog to th st of turl umbrs. For xmpl, R 3 (N) = :, b, c, d, N is th st of ll thr-dimsiol turl rhotrics. This st of turl rhotrix is butiful rhotrix with uiqu chrctristics my of which r yt to b discovrd. I this work, w will b lookig t som fudmtl proprtis of this lgbric st. Dfiitio.2 Ajibd (2003), Mohmmd d Tll (202) A rl rhotrix st of dimsio thr, dotd s R 3 (R) ws dfid by Ajibd s R 3 (R) = :, b, c, d, R whr h(r) =c is clld th hrt of y rhotrix R blogig to R 3 (R) d R is th st of rl umbrs. Exmpls showig xtsio of this st d lysis r copious i litrtur (Amiu & Michl, 205; Bumslg & Chdlr, 968; Ezugwu, Ajibd, & Mohmmd, 20; Mohmmd, 2009, 204; Mohmmd, Blrb, & Imm, 202; Tuduky & Mjuol, 200; Usii & Mohmmd, 202). It is worthy to ot tht -dimsiol rl hrt-bsd rhotrix dotd by R (R), will hv it crdility s R (R) = 2 (2 ), whr 2Z. This implis tht ll hrt-bsd or hrt-oritd rhotrics r of odd dimsio ( 3). All oprtios (dditio d multiplictio) i this work, will b s dfid by Ajibd i (2003). Thus, dditio d multiplictio of two hrt-bsd rhotrics r dfid s: R Q = d R Q = b h(r) d f g h(q) j k = h(q)fh(r) bh(q)gh(r) h(r)h(q) dh(q)jh(r) h(q)kh(r) f b g h(r)h(q) d j k rspctivly. A grliztio of this hrty multiplictio is giv i Mohmmd (204) d i Ezgwu t l. (20), Pg 2 of 0

3 A row-colum multiplictio of hrt-bsd rhotrics ws proposd by Si (2004) s: f dg R Q = bf g h(r)h(q) j dk bj k A grliztio of this row-colum multiplictio ws lso ltr giv by Si (2007) s: t t R Q = ij, c i2j2 b i2j2, d l2k2 = ( i j b i2 j ), (c 2 l k d l2 k ), t =( ) 2. 2 i 2 j = l 2 k = whr R d Q r -dimsiol rhotrics (with rows d colums). Dfiitio.3 A rhotrix is sid to b ivrtibl if it hs ivrs For xmpl lt A = b h(a) d Th, th ivrs of A is giv s A = (h(a)) 2 b h(a) d It is obsrvd tht y rhotrix h(r) 0 is ivrtibl or o-sigulr rhotrix. Howvr, turl rhotrix is sigulr rhotrix. Tht is, w cot fid A for which A is turl rhotrix. 2. Prlimiris I this sctio, w shll strictly cocr ourslvs with th turl rhotrix rhotrix whos prt st is th st of wll-ordrd turl umbrs. A turl rhotrix strts with dimsio o (i.. R ). Thrfor, th crdility of -dimsiol turl rhotrix is giv by R (N) = 2 (2 ), whr 2N. Rmrk 2. Rcll tht th st N ={0,, 2, 3, }. s Ashikplokhi, Agbboh, Uzor, Elkh, d Isr (200), Bumslg d Chdlr (968)-Th st of turl umbr or bttr still th st of ogtiv itgrs. This st 2N is lrgr th 2Z. Thrfor, this ppr is xpdig th scop of st of rhotric dimsios s hithrto prstd i litrtur. Dfiitio 2. (Mjor row d mjor colum) Th mjor row d th mjor colum r usully th oly full row d full colum i rhotrix. Thy r usully t th middl of th rows d colums of y dimsiol rhotrix. A turl rhotrix, s othr rhotrics, hs o mjor row d o mjor colum. 3. Exmpls of turl rhotrics This sctio givs diffrt rprsttios of turl rhotrics s xmpls ccordig to thir dimsios. () A turl rhotrix of dimsio o (R ) is giv by: R = whr N. It oly hs hrt with o othr tris. Pg 3 of 0

4 (b) A turl rhotrix of dimsio thr (R 3 ) is giv by: R 3 = whr, b, c, d, N Rmrk 3. All th tris r o-zro lmts of N. This rmrk holds for th othr xmpls of turl rhotrics. (c) A turl rhotrix of dimsio fiv (R 5 ) is giv by: R 5 = f g h j k l m whr, b, c, d,, f, g, h, j, k, l, m, N (d) A turl rhotrix of dimsio sv (R 7 ) is giv by: R 7 = f g h i j k l m o p q r s t u v w x y whr, b, c,, y N. Rmrk 3.2 You c go o d o. For xmpl, R 9 d R will hv thir lst tris s 4 d 6, rspctivly. () Grlly, turl rhotrix of dimsio (R ) is giv by: R = whr, b, c,,2 2 2 N 2N d N (i.. = 2 ) Rmrk 3.3 Th bov is th grliztio of y turl rhotrix. Pg 4 of 0

5 4. Proprtis of turl rhotrix Lmm 4. Lt R i b y i dimsiol turl rhotrix. Th, th hrt (h(r i )) is th middl vlu of st of umbrs tht mk up th rhotrix if d oly if = R i i =, 3, 5, Proof First prt Sic 2N, th thr xist mddl vlu (mdi). So, if = R i,, 3, 5, th for i = is trivil. So, i = 3 = 5 tris which r ordrd turl umbrs. Thus, th mdi is 3 = 2 ( R 3 ). So, i = 5 = 3 tris which r ordrd turl umbrs. Thus, th mdi is 7 = 2 ( R 5 ). So, i = 2k = 2k 2 2k tris which r ordrd turl umbrs. Thus, th mdi is 2 = 2 ( R 2k ). Th covrs follows from th Crdility of R whr 2N. Thorm 4. Lt R b y dimsiol turl rhotrix. Th, th followig r quivlt: () Th crdility R = 2 (2 ) whr 2N (b) Th lst try will b th vlu N (c) Th hrt of R [h(r )] is rprstd by h = ( R 2 ), 2N. (d) Th h(r ) will b th vlu 2 Proof () (b) Sic R = 2 (2 ) whr 2N, th for ll N, = 2. Th R( ) = (b) (c) Sic th lst try is d is old, th by Lmm 4., th middl vlu is (c) (d) Giv tht 2 = 2 ( R ) =h(r ) 2N h(r )= 2 ( R ) 2N d lttig = 2 givs h(r )= 2 (d) () Sic h(r )= 2 d by Lmm 4., 2h(R )= R, th R = 2 (2 ) Rmrk 4. Thorm 4. is simply chrctriztio of th turl rhotrix 4.. Dtrmit fuctio Though, th turl rhotrics r ot ivrtibl rhotrics. Howvr, th hrt of turl rhotrix plys th rol of dtrmit fuctio. Pg 5 of 0

6 Lmm 4.2 Lt A d B b y turl rhotrics of dimsio d A dtrmit fuctio of A, th AB = A B Proof Lt A = h(a) d B = h(b) th AB = h(a)h(b) = A B Rmrk 4.2 (i) Agi, th justpositio AB rprsts A B s dfid i Ajibd (2003) (ii) W will dopt h(a) to m th dtrmit fuctio of A. For xmpl, cosidr thr-dimsiol rhotrix (R 3 ): A = b h(a) d Th dtrmit fuctio of th bov rhotrix A is dsigtd by h(a) Computig th vlu of th dtrmit fuctio of y turl rhotrix is simply th vlu of its hrt. Howvr, for highr turl rhotrics, w itroduc th cocpt of mior rhotrics d codtrmit fuctios. Tht brigs us to th xt subsctio Codtrmit fuctio Th cocpt of codtrmit fuctios is similr to tht of mior mtrics. I turl rhotrics of dimsio thr (R 3 ), th codtrmit fuctio is th sm s th dtrmit fuctio. But for highr turl rhotrics th codtrmit fuctios r ot cssrily th sm. To vlut th codtrmit fuctio of highr turl rhotrix, w d to first split or rduc th highr turl rhotrix ito chi(s) of R 3 clld th mior rhotrics d th dtrmit fuctio of ch of thm is vlutd s bov. Ths mior rhotrics r split or rducd ithr log th mjor colum or log th mjor row d thir dtrmit fuctios r summd up ccordigly. For turl rhotrix with wll-ordrd tris, th rsult is th sm rgrdlss of whthr it is summd up log th mjor colum or log th mjor row. Cosidr th turl rhotrix of dimsio fiv blow: A = f g h j k l m Bhold, h(a) =g. To vlut codt(a), th highr rhotrix ds to b split ito mior rhotrics of dimsio thr. codt(a) = f g h j = c l k l m Tht is splittig th rhotrix A log th mjor colum givs: Pg 6 of 0

7 codt(a) = g g k l m = c l Whil splittig log th mjor row givs: codt(a) = b f g k Rmrk 4.3 For turl rhotrix c l = f h. d g h j m = f h Exmpl 4. Fid th dtrmit d th codtrmit fuctios of th turl rhotrix blow: A = Solutio h(a) =7 Nxt, w fid codtrmit fuctio, first log th mjor colum givs: codt(a) = Now log th mjor row givs: codt(a) = Extsio to highr dimsio c b md i similr mr usd i rducig R 5 to mior rhotrics of R 3. Lt us cosidr th xt xmpl. = 3 = 4 = 6 8 = 4 Exmpl 4.2 Fid th dtrmit d th codtrmit fuctios of th followig: (i) log mjor colum (ii) log th mjor row of th rhotrix blow: Q = Solutio h(q) =3 Pg 7 of 0

8 (i) codt(q) = (ii) codt(q) = = = 39 = 3 5 = 39 It is of utmost importc tht th rdr prctics vlutig th codtrmit fuctios of highr rhotrics log mjor row or mjor colum. This will giv th rdr th prrquisit cofidc i obtiig th vlus of codtrmit fuctios i crti css whr prticulr mjor row or colum tds to cclrt th rls of rsults which cosqutly lds to rducd hrdship d computtio tim xpdd. Cosidrig th Exmpl 4.2, summig log th colum sms to mk lif sir for th rdr. Howvr, th buty of this work lis i its simplicity Idx of rhotrics Th idx of turl rhotrix A is th umbr of mior rhotrics of dimsio thr tht c b drivd, ithr log th mjor colum or log th mjor row, from A. This idx is whol umbr or bttr still turl umbr. For xmpl th idx of R 3 is d of R 5, R 7, d R 9 r 2, 3, d 4, rspctivly. Appropritly, th idx of R is zro. Thorm 4.2 Giv y rhotrix R, th codt(r) =ρh(r) whr ρ is turl umbr clld th idx of R. Proof W prov usig mthmticl iductio. Sic idx of turl rhotrix is turl umbr corrspodig to th umbr of R 3 tht c b drivd from R, 3, d 2N. Now, wh = 3, th codt(r 3 )=h(r 3 ) sic th codt(r 3 ) is cssrily th dt(r 3 )=h(r 3 ). By Lmm 4.. Implis tht ρ =, So, th qutio is tru for = 3. For = 5, th w hv two miors of R 3. Tht is, codt(r 5 )=2h(R 5 ) implis tht ρ = 2. So, th qutio is tru for = 5. For = 7, th, w hv thr miors of R 3, th codt(r 7 )=3h(R 7 ) So, th qutio is tru for = 7 d ρ = 3. Th, for = 2k, codt(r 2k )=kh(r 2k ) Th, it is tru for = 2k d ρ = k. Pg 8 of 0

9 For = 2k 3, codt(r 2k3 )= codt(r 2(k) )=k h(r 2(k) ) Th, it is tru for = 2k 3 d ρ = k. Hc, th qutio is tru for ll vlu of 3 d ρ turl umbr. Thorm 4.3 Givig y turl rhotrix R codt(r) = ρ ( R ) 2 whr ρ is th idx d R is th crdility of R, d R = 2 (2 ) Proof Sic codt(r) =ρh(r) d by Lmm 4., dtrmit fuctio is h(r). Th, th rsult follows from th Thorm Coclusio This rticl xmid th proprtis of th turl rhotrix st, itroducd th cocpts of mior rhotrics, dtrmit fuctios, codtrmit fuctios d idx of turl rhotrics. Ths cocpts r ovlty to rhotrix lgbr, d thir mthods of computtios r prstd for mthmticl richmt. Usig th rsults i this ppr, o c vlut th dtrmit fuctio of y -dimsiol rhotrix t glc. With th liks btw th dtrmit fuctios, codtrmit fuctios d th idx o will b bl do sktch of y -dimsiol turl rhotrix o mttr how lrg is th vlu of. Ackowldgmts Th uthor wishs to xprss his profoud grtitud d pprcitio to Dr Abdul Mohmmd of Ahmdu Bllo Uivrsity, Zri for his courgmt d vlubl discussio, d to th vrious oymous rviwrs whos cotributios hv hlpd to improv this work. Fudig Th uthor rcivd o dirct fudig for this rsrch. Author dtils A.O. Isr E-mils: bdis@yhoo.co.uk, isro@ukpom.du.g Dprtmt of Mthmtics, Ambros Alli Uivrsity, Ekpom 3000, Nigri. Cittio iformtio Cit this rticl s: Nturl rhotrix, A.O. Isr, Cogt Mthmtics (206), 3: Rfrcs Ashikplokhi, U. S. U., Agbboh, G. U., Uzor, J. E., Elkh, A. O., & Isr, A. O. (200). A first cours i lir lgbr d lgbric struturs. PON Publishrs. Ajibd, A. O. (2003). Th cocpt of rhotrix i mthmticl richmt. Itrtiol Eductio i Scic d Tchology, 34, Amiu, A., & Michl, O. (205). A itroductio to th cocpt of prltrix, grliztio of rhotrix. Jourl of th Afric Mthmticl Uio d Sprigr-Vrlg, 26, Atssov, K. T., & Sho, A. G. (998). Mtrix-trtios d mtrix-oitrts: Exrcis for mthmticl richmt. Itrtiol Eductio i Scic d Tchology, 29, Bumslg, B., & Chdlr, B. (968). Thory d problms of group thory. Schum s outli sris. Ezugwu, E. A., Ajibd, A. O., & Mohmmd, A. (20). Grliztio of hrt-oritd rhotrix multiplictio d its lgorithm implmttio. Itrtiol Jourl of Computr Applictios, 3, 5. Mohmmd, A. (2009). A rmrk o th clssifictios of rhotrics s bstrct struturs. Itrtiol Jourl of Physicl Scics, 4, Mohmmd, A. (204). A w xprssio for rhotrix dvcs i lir lgbr & mtrix thory, 4, Mohmmd, A., Blrb, M., & Imm, A. T. (202). Rhotrix lir trsformtio. Advcs i Lir Algbr & Mtrix Thory, 2, Mohmmd, A., & Tll, Y. (202). Rhotrix sts d rhotrix spcs ctgory. Itrtiol Jourl of Mthmtics d Computtiol Mthods i Scic d Tchology, 2, 202. Si, B. (2004). A ltrtiv mthod for multiplictio of rhotrics. Itrtiol Jourl of Mthmticl Eductio i Scic d Tchology, 35, Si, B. (2007). Th row-colum multiplictio for highr dimsiol rhotrics. Itrtiol Jourl of Mthmticl Eductio i Scic d Tchology, 38, Si, B. (2008). Covrsio of rhotrix to coupld mtrix. Itrtiol Jourl of Mthmticl Eductio i Scic d Tchology, 39, Tuduky, S. M., & Mjuol, S. O. (200). Rhotrics d th costructio of fiit filds. Bullti of Pur d Applid Scics, 29, Usii, S., & Mohmmd, L. (202). O th rhotrix igvlus d igvctors. Jourl of th Afric Mthmticl Uio d Sprigr-Vrlg, 25, Pg 9 of 0

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