Solution of Linearly-Dependent Equations by Generalized Inverse of a Matrix

Transcription

1 It. J Sci. Eergig ech Vol No ugust Solutio of iearlydepedet Equatios y eeralied Iverse of a Matri wi M. usaa ssociate Professor of pplied Matheatics Matheatics Departetlahar Uiversityaa P.O.Bo 77, aaaa Strip, Palestiia uthority 8 stracthis paper riefly reviews the atheatical cosideratios ehid the geeralied iverse of a atri. Siple derivatios for the deteriatio of differet types of geeralied iverses of a atri are preseted. hese iclude results of the geeralied iverse of sigular ad rectagular atrices. It also icludes applicatios of the geeralied iverse to solutio of a set of liearly depedet equatios. Keywords: eeralied Iverse of a atri, Sigular atri, iearly Depedet Equatios, Matheatical software. MS Suject Classificatio: 9. Itroductio Sigular ad early sigular atrices as well as atrices with large coditio uers are oipreset i coputatios for physics, egieerig, edicie, sigal processig, cotrol theory, filterig, statistics ad codig theory [7]. epresetatios of such a atri eale its fast ultiplicatio y a vector ad epressio of its iverse via the solutios of a few liear systes of equatios. he later proles of iversio ad liear syste solvig are highly iportat for the theory ad practice of coputig, the realtie iversio of atrices is usually desired. Sice the id98s, efforts have ee directed toward coputatioal aspects of fast atri iversio, ad ay algoriths have thus ee proposed. he cocept of a geeralied iverse was first itroduced y Fredhol 9, he called a particular geeralied iverse as Pseudo iverse which serve as itegral operator. he class of all Pseudo iverses was characteried i 9 y Hurwit who used fiite diesioality of ull operators of Fredhol operators to give a siple algeraic costructio. eeralied iverse of differetial operators, already iplicit i Hilert s discussios i 9 of geeralied ree fuctios were cosequetly studied y uerous authors, i particular, Myller i 9, Bouitky i 99, Elliott i 98 ad eid i 9. Iteratioal Joural of Sciece & Eergig echologies IJSE, EISSN: 8888 Copyright Eceligech, Pu, UK However, he cocept of a iverse of a sigular atri sees to have ee first itroduced y Moore [, ] i 9. Etesios of these ideas to geeral operators have ee ade y seg [8], ut o systeatic study of the suject was ade util 9 whe Perose [], redefied the Moore iverse i slightly differet way. he ethod of coputig what is called a pseudoiverse of a sigular atri was discussed y ao [], he also applied it to solve oral equatios with sigular atri i the least squares theory ad to epress the variaces of estiators. he geeralied iverse cocept was discussed y ao [, 7] for the weaker defiitio, he showed that they are ot uique ad thus presets a iterestig study i atri algera. He showed how a variety of geeralied iverses could e costructed to suit differet purposes ad preseted a classificatio of the geeralied iverses. eeralied iverse is a great tool i solvig liearly depedet ad ualaced syste of liear equatios. It has the aility to fid the solutio of square ad osquare atrices eve whe they are sigular. he coputatio of the socalled ero iitial state syste iverses for liear tieivariat state space syste is essetially equivalet to deteriig geeralied iverses of the associated trasferfuctio atrices. I this paper we discuss a calculus of geeralied iverses ad show how it provides a elegat tool for the discussio of fidig the solutio of liearlydepedet equatios prole. we address uerically reliale coputatio of geeralied iverses of sigular atrices. We preset a uerical algorith for coputig the solutio of liearlydepedet equatios, ad give soe uerical eaples to illustrate our ethods, ad report o MB software [] to copute soe geeralied iverses.. eeralied Iverses atri. he a atural questio et e is whe we ca solve the syste

2 It. J Sci. Eergig ech Vol No ugust 9 for, give. If is a square atri ad has a iverse, the. holds if ad oly if. his gives a coplete aswer if is ivertile. However, ay e with, or ay e square atri that is ot ivertile. If is ot ivertile, the equatio. ay have o solutios that is, ay e ot i the rage of, ad if there are solutios, the there ay e ay differet solutios. For eaple, assue.he, so that is ot ivertile. It would e useful to have a characteriatio of those for which it is possile to fid a solutio of, ad, if has a solutio, to fid all possile solutios. It is easy to aswer these questios directly for a atri, ut ot if were 8 or. solutio of these equatios ca e foud i geeral fro the otio of a geeralied iverse of a atri: Defiitio. If is a a geeralied iverse of if is a atri with atri, the is. d of course if is a geeralied atri the is a solutio of. If has a iverse i the usual sese, that is if is has a twosided iverse hus if, the ad,while. eists i the usual sese, the. his of course justifies the ter geeralied iverse. eeralied iverse is of great iportace i solvig liearly depedet ad ualaced equatios; eeralied iverse has a lot of applicatios to osquare ad square sigular atrices. If is a osigular atri, the there eists a. uique iverse I with the property. he fact that has a geeralied iverse, eve whe it is sigular or rectagular has iportat applicatios i the prole of solvig syste of equatios of the for. If is rectagular atri with rak the eists, ad defiig we fid that I. I such a case is called a left iverse of. Siilarly a right iverse of eists if its rak is with the property I. Whe,, or eists we ca epress a solutio of the equatio i the for or, or. Whe such iverses do ot eist, ca we preset a solutio of the cosistet equatio where ay e rectagular or a square sigular i the for? If such a eists, we call it a geeralied iverse of. y atri has at least oe geeralied iverse. However, uless is ad is ivertile, there are ay differet geeralied iverses, so that geerally is ot uique, ad they are uique oly if you ipose ore coditios o. y atri has at least oe geeralied iverse. Sice ofte ay differet liearlyidepedet sets of r rows ca e peruted to the upper r rows ad ay differet liearlyidepedet sets of r colus ca e peruted ito the first r colu positios, a atri with rak r < ca have ay differet geeralied iverses. I geeral, a square atri P that satisfies P P is called a projectio atri. Oe cosequece of. is that ad. hus oth ad are projectio atrices. Sice is ad is, is a projectio atri ad is. he followig theore shows that, the two projectios ad ca e used to solve atri equatios. atri ad heore.. et e a assue that is a geeralied iverse of. he, for ay fied, i he equatio, Has a solutio if ad oly if

3 It. J Sci. Eergig ech Vol No ugust ii If has ay solutios, the is a solutio of if ad oly if + I for soe particular solutio of for i the rage of ca e otaied fro. Nuerical Coputatios s a eas of otivatig a siple uerical eaples of how the geeralied iverse works, we develop a rief applicatio to the liear odel where the atri where the atri is oivertile sice it is sigular. I this cotet, the geeralied iverse provides a solutio to the oral equatios. he followig algorith is ipleeted ad siulated for MB [] user suroutie for solvig a syste of liearlydepedet equatios. lgorith for the eeralied Iverse ad solutio of ssue is atri ad a colu vector Choose ay osigular suatri H of diesio r, Fid H, eplace the eleets of suatri H i the origial atri y eleets of H, eplace all other eleets y eros to get a ew atri, he geeralied atri, Calculate Use + I solutio of, to calculate a Eaple.. et he., Set.. So is a geeralied iverse of. he two. projectios ad ca e used for solvig a syste for i the rage of I this case y + y + y y + y hus has a solutio c, o the other had, y. y oly if y So that the rage of projectio is eactly the set of vectors c ccordig to the theore if of solutios of is eactly c + I. c +. c +. c., the the set Eaple.. Cosider the syste of cosistet equatios with:, ad H

4 It. J Sci. Eergig ech Vol No ugust So H is a geeralied iverse atri of. he solutio of the syste is ets cosider aother choice of H : H So the ew is also a geeralied iverse of the atri. It iplies that the geeralied iverse of a atri is ot uique, it depeds o the uer of otaied osigular atrices H of rak, ad for the a give the solutio of is ot uique. he other solutio of is give y I For, we otai the first solutio of the syste, ad for.98 the other solutio is otaied, it iplies that, all solutios of the differet geeralied iverses ca e otaied fro oe geeralied iverse. Eaple.. et us cosider the followig atri Note that rak H Coclusios We have discussed uerically reliale ethods ad coputer algorith to copute geeralied iverses of sigular atrices. he proposed ethods are copletely geeral, eig applicale to sigular atrices. he proposed approach provides fleiility to copute the solutios of liearlydepedet equatios, it has ee also show that all solutios

5 It. J Sci. Eergig ech Vol No ugust ca e otaied fro oly oe geeralied iverse atri. efereces [] di BeIsrael ad. E. reville, eeralied iverses: heory ad pplicatios. d Ed. New York, NY: Spriger,. ISBN [] E. H. Moore, eeral alysis, Philadelphia, erica Philosophical Society, 9. [] E. H. Moore, O the eciprocal of the eeral lgeraic Matri astract, Bull. er. Math. Soc., Vol. 9, pp [] MB User s uide, Versio 7., he Math Works Ic.,. []. Perose, geeralied Iverse of Matrices, Proc. Caridge Philos. Soc., Vol. 9, pp.. [] C.. ao, alysis of Dispersio for Multiply Classified Data with Uequal Nuers i Cells, Śakhyä. Vol. 9, pp. 8. [7] C.. ao ad S. K. Maitra, eeralied Iverse of Matrices ad It s pplicatios, Wiley, New York, 97 [8] Y. Y. seg, eeralied Iverses of Uouded Operators Betwee wo Uitary Spaces, Dokl. kad. Nauk. SSS., Vol. 7 99, pp.. [9] Y. Y. seg, Properties ad Classificatios of eeralied Iverses of Closed Operators, Dokl. kad. Nauk. SSS, Vol. 7 99, pp. 7. [] Y. Y. seg, Virtual Solutios ad eeral Iversios, Uspehi Mat. Nauk., Vol. 9, pp.. [] B. Zheg ad. B. Bapat, eeralied Iverse, S ad a ak Equatio, pplied Matheatics ad Coputatio Vol., pp. 7.