ON THE MATRICES KEEPING INVARIANT THE QUADRATIC FORMS

Transcription

1 UPB Sci Bull Series A Vol 7 Iss 4 ISSN -77 ON HE MARICES KEEPING INVARIAN HE QUADRAIC FORMS Alia-PERESCU-NIŢĂ Carmia GEORGESCU Î această lucrare se studiaă matricele care ivariaă o formă pătratică reală fiată şi se dă u criteriu ca acestea să formee u grup; î se dau câteva eemple Î se studiaă grupul matricelor care ivariaă forma pitagoreică şi se dă u procedeu de obţiere sistematică a soluţiilor ecuaţiei diofatice I this work oe aales coditios that a real quadratic form is kept ivariat b some matrices I some eamples are give ad i oe studies the group of matrice which keep ivariat the pitagoreic form ad oe idicates a procedure to get sstematicall the solutios of the diophatic equatio Kewords: Quadratic form ivariat group of matrices a pitagoreic group Itroductio Quadratic forms are importat geometrical algebraic objects both for the stud of various mathematical structures but also for their applicatios to the stud of quadrics i matri calculus differetial geometr ad tesor calculatio A geeral theoretical presetatio is foud i [] ad more specific problemes are treated i [] [] Usig [4] the paper deals with the stud of matrices which preserve a real give quadratic form I the secod part of the paper the results are applied to the sstematic listig of Pthagorea triples of umbers O the coectio betwee matrices ad quadratic forms Lecturer Depart of Mathematics Uiversit POLIEHNICA of Bucharest Romaia ita_alia@ahoocom Lecturer INRIA Rhoe Aples Iovall ee 655 aveue de l Europe 8 Motboot Frace carmiageorgescu@irialpesfr

2 46 Alia Petrescu-Niţă Carmia Georgescu O the coectio betwee matrices ad quadratic forms Fi a real quadratic form q : R R B defiitio a matri A M R keeps ivariat the quadratic form q if the followig coditio holds: for a q we will idetif ad its traspose We will deote b G q the set of all matrices from M R which keep ivariat q If Q M R is the matri associated to q for the caoical basis of the space R it is well kow the Q is smmetric ad moreover q Q for a R Propositio he coditio is equivalet to A Q A Q Proof We have q Q ad q Q A Q he relatio becomes Q A Q A for a R whece Propositio he set G q is stable to products Proof Let A B G q We have to prove that B A Gq But for a R q ad q B ake hece q B B for a R I other terms B A G q Corollar G q GL R is a group with respect to multiplicatio of matrices Proof Obviousl the uit matri I belogs to G q If A Gq is osigular we will show that A G q Ideed for a R fied take A B q hece q A A therefore A G q NOE Suppose that ad q he matri A verifies hece belogs to G q but A is sigular

3 O the matrices keepig ivariat the quadratic forms 47 Propositio If q is odegeerated the G q is a group relative to multiplicatio Proof B defiitio q has all the proper values of the matri Q associated to q i the caoical basis are differet from ero hece det Q It is eough to show that a matri A Gq is osigular; ideed from the relatio it follows that det Q det Qdet A whece det A herefore det A ad A is osigular It remais to appl the corollar of the Propositio Eamples ake q I this case Q I ad the coditio becomes: A A I hus A is a orthogoal matri of order I the case oe obtai the rotatio matrices Cosider the case ad q a b I this case Q ad if A c d keeps ivariat q the from oe gets: a b c d ad a c ab cd he a b ad c d If b the k d b a c a c hece k herefore k or k d b d b a b a b So A or A with a b b a b a If b the a d ad we refid the above matrices I the case 4 ad q 4 4 which is odegeerated the the set G q has a structure of group b propositio which ca be called the group of Loret trasformatios of the 4-dimesioal space of Eistei-Mikowski Other results are i [] he matrices which keep ivariat the quadratic form q : R R q I this case Q diag ad b the propositio t G q { A M R A Q A Q}

4 48 Alia Petrescu-Niţă Carmia Georgescu is a group relativel to multiplicatio sice q is odegeerated If A Gq the det A ; whece det A ± ad particularl A is a b c osigular If A the b it follows the sstem: u v w a u b v c w ab uv bc vw ca wu Propositio 4 For q the set P G q M Z is a group relativel to multiplicatio Proof Obviousl if A B P the A B P b the propositio It remais to prove that a matri A P has a iverse i P Ideed sice * * det A ± it follows that A A ± A hece A has all elemets i det A Z ad thus A P Defiitio B obvious reaso the group P ca be called the pitagoreic group coected b the diophatic equatio Wheever a -uple is a solutio of the equatio the q ad b the -uple A is also a solutio of the same equatio for a A P Cosider the matrices Q diag R diag ad S he Q R S I hece Q Q R R S S Deote b P the subgroup of P geerated b Q R S hece p q r P { Q R S p q r { }} B eplicit calculus P cosists of the followig 6 matrices: m diag I m diag Q m diag m4 diag m 5 diag R m6 diag m 7 diag m8 diag

5 O the matrices keepig ivariat the quadratic forms 49 9 m S m m m m 4 m 5 m 6 m heorem a P is formed b the matrices of the form diag c b a or c b a where } { c b a b P P Proof a Represets the sthesis of the above eplicitatio b We have to idicate a matri \ P P Cosider the algebraic idetitites q q 4 ad the liear map : R R t t where are give b the relatios 5 whece 4 q q b b hus qt q for a therefore the matri associated to the liear map t i the caoical basis of R belogs to P O the other had

6 5 Alia Petrescu-Niţă Carmia Georgescu solvig the liear sstem it follows that hece Sice mk for a k 6 it follows that P he matri is ivertible with the iverse ad moreover q for a triple We also cosider the matrices V m 5 ad Z W m hese matrices V W belog to the group G q Defiitio wo triples of pitagoreic umbers u v ' ' ' ie o-ull solutios i N of the equatio are said equivalet if * there is N such that either v u or u v For istace ad 45 are ot equivalet but 45 ad 68 are equivalet heorem Let s 45 be the stadard solutio of the equatio * i N he set S of all oequivalet solutios i N of the equatio is just the set of the vertices of the followig ordered tree:

7 O the matrices keepig ivariat the quadratic forms 5 s s Vs Ws s Vs Ws Vs V s WVs Ws VWs W s I other terms all the elemets of S are obtaied b startig with s ad multiplig it with V W take i a arbitrar order Proof If P put p q r that is p q ad r Oe easil verifies that < r < ad p q caot be both egative If S ad ' ' ' the ' ' hece is preserved; the ' ' ad ' ' hece ad icrease O the other had if '' '' '' V the '' '' hece icreases ' ' ' ' hece is preserved ad '' '' that is icreases Fiall if ''' ''' ''' W oe gets ''' '' ' ''' '' ' ad ''' ''' hece icrease ad is preserved hus startid from s all the pitagoreic triples a b c such that a b are obtaied b s to he pitagoreic triples a b c such that c b or c a correspod V s ad those havig compoets that differ b are obtaied from W s Oe ca eplicit the powers V W b diagoalisatio of V W For istace oe gets that V s ad

8 5 Alia Petrescu-Niţă Carmia Georgescu W s which correspod to the solutios of the form a b a ad respectivel a b b from the set S 4 Coclusios he mai aim of the work is to stud the group of all matrices which keep ivariat the pitagoreic quadratic form Oe kows the classical parametriatio of the set S of all triples of o-ull atural umbers such that ; amel k p q pqk k p q with * k p q N p q he theorem allows aother listig of the elemets of S At a level r of the above ordered tree oe get r solutios Eplicitl startig with s are obtai 45 V W R E F E R E N C E S [] Alia Niţă Geeralied iverse of a matri with applicatios to optimiatio of some sstems PhD hesis i Romaia Uiv of Bucharest Facult of Math ad Comp Sciece 4 [] R Godemet Cours d algebre Herma 978 [] K Kitacha Arithmetic of quadratic forms Cambridge Uiversit Press 999 [4] Z Borevici IR Safarevici eoria umerelor Number theor EdStiitifica si Eciclopedica 985 i Romaia [5] DZ Djokovic S Severii Ratioal orthogaal versus real orthogoal J of Algebra