6 Determinnts One person s onstnt is nother person s vrile. Susn Gerhrt While the previous hpters h their ous on the explortion o the logi n struturl properties o projetive plnes this hpter will ous on the question: wht is the esiest wy to lulte with geometry?. Mny textooks on geometry introue oorintes (or homogeneous oorintes) n se ll nlyti omputtions on lultions on this level. An nlyti proo o geometri theorem is rrie out in the prmeter spe. For ierent prmeteriztion o the theorem the proo my look entirely ierent. In this hpter we will see, tht this wy o thinking is very oten not the most eonomi one. The reson or this is tht the oorintes o geometri ojet re in wy sis epenent rtit n rry not only inormtion on the geometri ojet ut lso on the reltion o this ojet with respet to the sis o the oorinte system. For instne i point is represente y its homogeneous oorintes (x, y, z) we hve enoe its reltive position to rme o reerene. From the perspetive o projetive geometry the perhps most importnt t tht one n sy out the point is simply the t tht it is point. All other properties re not projetively invrint. Similrly i we onsier three points p =(x,y,z ), p 2 =(x 2,y 2,z 2 ), p 3 =(x 3,y 3,z 3 ), the sttement tht these points re olliner res: x y 2 z 3 + x 2 y 3 z + x 3 y z 2 x y 3 z 2 + x 2 y z 3 + x 3 y 2 z = 0, 3 3 eterminnt. Agin rom struturl point o view this expression is y r too omplite. It woul e muh etter to enoe the ollinerity iretly into short lgeri expression n el with this. The simplest wy to o this is to hnge the role o primry n erive lgeri ojets.
00 6 Determinnts I we onsier the eterminnts themselves s irst lss itizens the sttement o ollinerity simply res et(p,p 2,p 3 ) = 0 where the eterminnt is onsiere s n unrekle unit rther thn just shorthn or the ove expne ormul. In this hpter we will explore the roles eterminnts ply within projetive geometry. 6. A eterminntl point o view Beore we strt with the tretment o eterminnts on more generl level we will review n emphsize the role o eterminnts in topis we hve trete so r. One o our irst enounters o eterminnts ws when we expresse the ollinerity o points in homogeneous oorintes. Three points p,p 2,p 3 re olliner in RP 2 i n only i et(p,p 2,p 3 ) = 0. One n interpret this t either geometrilly (i p,p 2,p 3 re olliner, then the orresponing vetors o homogenous oorintes re oplnr) or lgerilly (i p,p 2,p 3 re olliner, then the system o liner equtions p,l = p 2,l = p 3,l =0 hs non-trivil solution l (0, 0, 0)). Dully we n sy tht the eterminnt o three lines l,l 2,l 3 vnishes i n only i these lines hve point in ommon (this point my e t ininity). A seon instne where eterminnts plye less ovious role ourre when we lulte the join o two points p n q y the ross-prout p q. We will give n lgeri interprettion o this. I (x, y, z) is ny point on the line p q then it stisies et p p 2 p 3 q q 2 q 3 =0. x y z I we evelop the eterminnt y the lst row, we n rewrite this s ( ) ( ) ( ) p2 p et 3 p p x et 3 p p y + et 2 z =0. q 2 q 3 q q 3 q q 2 Or expresse s slr prout ( ) p2 p (et 3, et q 2 q 3 ( ) p p 3, et q q 3 ( ) p p 2 ), (x, y, z) =0. q q 2 We n geometrilly reinterpret this eqution y sying tht ( ) ( ) ( ) p2 p (et 3 p p, et 3 p p, et 2 ). q 2 q 3 q q 3 q q 2 must e the homogeneous oorintes o the line l through p n q sine every vetor (x, y, z) on this line stisies l, (x, y, z) = 0. This vetor is nothing else thn the ross-prout p q.
6.2 A ew useul ormuls 0 A thir sitution in whih eterminnts plye unmentl role ws in the einition o ross rtios. Cross rtios were eine s the prout o two eterminnts ivie y the prout o two other eterminnts. We will lter on see tht ll three irumstnes esrie here will hve nie n interesting generliztions: In projetive -spe oplnrity will e expresse s the vnishing o ( + ) ( + ) eterminnt. In projetive -spe joins n meets will e niely expresse s vetors o su-eterminnts. Projetive invrints n e expresse s ertin rtionl untions o eterminnts. 6.2 A ew useul ormuls We will now see how we n trnslte geometri onstrutions into expressions tht only involve eterminnts n se points o the onstrution. Sine rom now on we will hve to el with mny eterminnts t the sme time we irst introue useul revition. For three points p, q, r RP 2 we set: [p, q, r] := et p p 2 p 3 q q 2 q 3. r r 2 r 3 Similrly, we set or two points in RP ( ) p p [p, q] := et 2. q q 2 We ll n expression o the orm [...]rket. Here re ew unmentl n useul properties o 3 3 eterminnts: Alternting sign-hnges: [p, q, r] = [q, r, p] = [r, p, q] = [p, r, q] = [r, q, p] = [q, p, r]. Linerity (in every row n olumn): Plükers Formul: [λ p + µ p 2, q, r] =λ [p, q, r]+µ [p 2, q, r]. [p, q, r] = p, q r. The lst ormul n e onsiere s shorthn or our evelopments on ross prouts, slr prouts n eterminnts in the previous setion.
02 6 Determinnts l 2 q l 2 q Fig. 6.. Two pplitions o Plükers µ. 6.3 Plüker s µ We now introue very useul trik with whih one n erive ormuls or geometri ojets tht shoul simultneously stisy severl onstrints. The trik ws requently use y Plüker n is sometimes lle Plüker s µ. Imgine you hve n eqution : R R whose zero-set esries geometri ojet. For instne, think o line eqution (x, y, z) x+ y + z or irle eqution in the plne (x, y) (x ) 2 +(y ) 2 r 2. Oten one is intereste in ojets tht shre intersetion points with given reerene ojet n in ition pss trough thir ojet. I the liner omintion λ (p) +µ g(p) gin esries n ojet o the sme type then one n pply Plükers µ. All ojets esrie y p λ (p)+µ g(p) will pss through the ommon zeros o n g. This is ovious sine whenever (p) = 0 n g(p) = 0 ny liner omintion is lso 0. I one in ition wnts to hve the ojet pss through speii point q then the liner omintion p g(q) (p) (q) g(p) is the esire eqution. To see this simply plugin the point q into the eqution. Then one gets g(q) (q) (q) g(q) = 0. With this trik we n very esily esrie the homogeneous o line l tht psses through the intersetion o two other lines l n l 2 n through thir point q y l 2,q l l,q l. Testing whether this line psses through q yiels: l 2,q l l,q l,q = l 2,q l,q l,q l 2,q =0 whih is oviously true. Lter on we will mke requent use o this trik whenever we nee st n elegnt wy to lulte speii geometri
6.3 Plüker s µ 03 [,, ] [,, ] [,, ][e,, ] [,, ][e,, ] = 0 e Fig. 6.2. Conitions or lines meeting in point. ojet. We will now use Plükers µ to lulte intersetions o lines spnne y points. Wht is the intersetion the two lines spnne y the point pirs (, ) n (, ). On the one hn the point hs to e on the line thus it must e o the orm λ + µ it hs lso to e on hene it must e o the orm ψ + φ. Ientiying these two expressions n resolving or λ, µ, ψ n φ woul e one possiility to solve the prolem. But we n iretly re o the right prmeters y using ( ul version o) Plükers µ. The property tht enoes tht point p is on the line is simply [,, p] = 0. Thus we immeitely otin tht the point: [,, ] [,, ] must e the esire intersetion. This point is oviously on n it is on sine we hve: [,, [,, ] [,, ] ] = [,, ] [,, ] [,, ] [,, ] = 0. We oul equivlently hve pplie the lultion with the roles o n interhnge. Then we n express the sme point s: [,, ] [,, ] In t, it is not surprise tht these two expressions en up t ientil points. We will lter on in Setion 6.5 see tht this is just reormultion o the well-known Crmer s rule or solving systems o liner equtions. How n we express the onition tht three lines,, e meet in point? For this we simply hve to test whether the intersetion p o n, is on e. We n o this y testing whether the eterminnt o these three points is zero. Plugging in the ormul or p we get: [e,, [,, ] [,, ] ] = 0. Ater expnsion y multilinerity we otin:
04 6 Determinnts [,, ][e,, ] [,, ][e,, ] = 0. This is the lgeri onition or the three lines meeting in point. At this time it is worthy to step k n mke ew oservtions: The irst n most importnt oservtion is tht we oul write suh projetive onition s polynomil o eterminnts evluting to zero. In the ormul eh term hs the sme numer o eterminnts. Eh letter ours the sme numer o times in eh term. All three oservtions generlize very well to muh more generl ses. However or this we will hve irst to introue the notion o projetively invrint properties. However, eore we o this we wnt to use this ormul to otin (nother) eutiul proo or Pppus s Theorem. Consier the rwing o Pppos s Theorem in Figure (oserve the nie 3-ol symmetry). We n stte Pppos s theorem in the ollowing wy: i or six points,..., in the projetive plne the lines,, e meet n the lines, e, meet then lso e,, meet. The two hypotheses n e expresse s [,, e][,, ] = [,, ][,, e] [,, ][, e, ] = [,, ][, e, ] By using the t tht yli shit o the points o 3 3 rket oes not hnge its sign, we oserve tht the irst term o the seon eqution is ientil to the seon term o the irst eqution. So we otin: [,, ][e,, ] = [,, ][e,, ] Whih is extly the esire onlusion o Pppos s Theorem. 6.4 Invrint properties How n we lgerilly hrterize tht ertin property o point onigurtion is invrint uner projetive trnsormtions? Properties o suh type re or instne three lines eing olliner or six points,..., suh tht,, e meet. In generl properties o this type n e expresse s untions in the (homogeneous) oorintes o the points tht hve to e zero, when the property hols. Being invrint mens tht property hols or point onigurtion P i n only i it lso hols or ny projetive trnsormtion o P. More ormlly, let us express the point onigurtion P y mtrix whose olumns re the homogeneous oorintes o n points p,...,p n. P = p x p 2x... p nx p y p 2y... p ny p z p 2z... p nz
6.4 Invrint properties 05 e Fig. 6.3. Pppos s Theorem, one more. A projetive trnsormtion is then simply represente y let-multiplition with 3 3 invertile mtrix T. A projetively invrint property shoul lso e invrint when we reple vetor p i y sler multiple λ i p i. We n express the sling o the points y right multiplition o P with n invertile igonl mtrix D. All mtries otine rom P vi T P D represent essentilly the sme projetive onigurtion. A projetive invrint property is ny property o P tht stys invrint uner suh trnsormtion. Very oten our invrint properties will e polynomils eing zero, ut or now we wnt to sty on more generl sie n onsier ny mp tht ssoites to P oolen vlue. The mtrix P n e onsiere s n element o R 3 n. Thus we eine Deinition 6.. A projetively invrint property o n points in the rel projetive plne is mp : R 3 n {true, lse}, suh tht or ll invertile rel 3 3 mtries T GL(R, 3) n n n invertile rel igonl mtries D ig(r,n) we hve: (P )=(T P D). In nonil wy we n ientiy eh preite P X on spe X with with its hrteristi untion : X {true, lse} where (x) evlutes to true i n only i x P. Thus we n equivlently spek o projetively invrint preites. In this sense or instne the [,, ] = 0 eines projetively invrint property on or three points,, in the rel projetive plne. Also the property tht we enountere in the lst setion
06 6 Determinnts [,, ][e,, ] [,, ][e,, ] = 0 whih enoe the t tht three lines,, e, meet in point is projetively invrint. Beore we stte more generl theorem we will nlyze why this reltion is invrint rom n lgeri point o view. Trnsorming the points y projetive trnsormtion T results in repling the points,..., with T,..., T. Sling the homogeneous oorintes results in repling,..., y λ,..., λ with non-zero λs. Thus i P enoes the points onigurtion, then the overll eet o T P D on the expression [,, ][e,, ] [,, ][e,, ] n e written s [λ T, λ T, λ T ][λ e T e, λ T, λ T ] [λ T, λ T, λ T ][λ e T e, λ T, λ T ]. Sine [T p, T q, T r] = et(t ) [p, q, r] the ove expression simpliies to (et(t ) 2 λ λ λ λ λ e λ ) ([,, ][e,, ] [,, ][e,, ]). All λs were non-zero n T ws ssume to e invertile. Hene the expression [,, ][e,, ] [,, ][e,, ] is zero i n only i the ove expression is zero. Oserve tht it ws importnt tht eh summn o the rket polynomil h extly the sme numer o rkets. This me it possile to tor out tor et(t ) 2. Furthermore in eh summn eh letter ourre the sme numer o times. This me it possile to tor out the λs. This exmple is speil se o muh more generl t, nmely tht eh multihomogeneous rket polynomils eine projetively invrint properties. Deinition 6.2. Let P =(p,p 2,,..., p n ) (R 3 ) n represent point onigurtion o n points. A rket monomil on P is n expression o the orm [,,,2,,3 ][ 2,, 2,2, 2,3 ]... [ k,, k,2, k,3 ] where eh i,j is one o the points p i. The egree eg(p i,m) o point p i in monomil is the totl numer o ourrenes o p i in M. A rket polynomil on P is sum o rket monomils on P. A rket polynomil Q = M +...+ M l with monomils M,..., M l is multihomogeneous i or eh point p i we hve eg(p i,m )=...= eg(p i,m l ). In other wors, rket polynomil is multihomogeneous i eh point ours in eh summn the sme numer o times. We n mke n nlogous einition or points on projetive line. The only ierene there is tht we hve to el with rkets o length 2 inste o length 3. As stright orwr generliztion o our oservtions on the multihomogeneous rket polynomil [,, ][e,, ] [,, ][e,, ] we otin:
6.4 Invrint properties 07 Theorem 6.. Let Q(P ) e multihomogeneous rket polynomil on n points P =(p,p 2,,..., p n ) (R 3 ) n then Q(P ) = 0 eines projetively invrint property. Proo. Sine Q is multihomogeneous eh o the summns ontins the sme numer (sy 3k) o points. Thereore eh summn is the prout o k rkets. Thus we hve or ny projetive Trnsormtion T the reltion Q(T P ) = et(t ) k Q(P ). Furthermore the egree o the point p i is the sme (sy i ) in eh monomil. Sling the points y slrs λ,..., λ n e expresse s multiplition with the igonl mtrix D = ig(λ,..., λ n ). Sine eh rket is liner in eh point-entry the sling inues the ollowing tion on Q: Overll we otin: Q(P D) =λ Q(T P D) = et(t ) k λ... λn n Q(P ).... λn n Q(P ). The tors preeing Q(P ) re ll non zero sine T is invertile n only non-zero λ i re llowe. Hene Q(T P D) is zero i n only o Q(P ) is zero. Clerly, similr sttement oes lso hol or points on the projetive line (n 2 2 rkets) n lso or projetive plnes over other iels. We oul now strt omprehensive stuy o multihomogeneous rket polynomils n the projetive invrints enoe y them. We will enounter severl o them lter in the ook. Here we just give without urther proos ew exmples to exempliy the expressive power o multihomogeneous rket polynomils. We egin with ew exmples on the projetive line [] = 0 oinies with = [][] + [][] = 0 (, ); (, ) is hrmoni [e][][] [][][e] = 0 (, ); (, ); (e, ) is qurilterl set e Here re some other exmples in the projetive plne
08 6 Determinnts [] = 0,, re olliner [][e] + [e][] = 0 The line pirs (, ); (, e) re hrmoni F e [e][] [][e] = 0 ( ); ( ); (e ) meet in point e [][e][][e] [e][][e][] = 0,,,, e, re on oni e 6.5 Grssmnn-Plüker Reltions When we stuie the exmple o three lines,, e meeting in point we ene up with the ormul [,, ][e,, ] [,, ][e,, ] = 0. A loser look t this ormul shows tht the line plys speil role ompre to the other two lines. Its points re istriute over the rkets, while the points o the other lines lwys our in one rket. The symmetry o the originl property implies tht there re two more essentilly ierent wys to enoe the property in rket polynomil. [,, ][e,, ] [,, ][e,, ] = 0 or [,, e][,, ] [,, e][,, ] = 0. The reson or this is tht there re multihomogeneous rket polynomils tht will evlute lwys to zero no mtter where the points o the onigurtion re ple. These speil polynomils re o unmentl importne whenever one mkes lultions where severl eterminnts re involve. The reltions in question re the so lle Grssmnn-Plüker-reltions. In priniple, Suh reltions exist in ny imension. However, s usul in our exposition we will minly ous on the se o the projetive line n the projetive plne, i.e. 2 2 n 3 3 rkets. We strt with the 2 2 se. We stte the reltions on the level o vetors rther thn on the level o projetive points. Theorem 6.2. For ny vetors,,, R 2 the ollowing eqution hols: [, ][, ] [, ][, ] + [, ][, ] = 0
6.5 Grssmnn-Plüker Reltions 09 Proo. I one o the vetors is the zero vetor the eqution is trivilly true. Thus we my ssume tht eh o the vetors represents point o the projetive line. Sine [, ][, ] [, ][, ] + [, ][, ] is multihomogeneous rket polynomil we my ssume tht ll vetors re (i neessry ter suitle projetive trnsormtion) inite points n normlize to vetors ( λ ) (,..., λ the term gives: ). The eterminnts then eome simply ierenes. Rewriting (λ λ )(λ λ ) (λ λ )(λ λ ) + (λ λ )(λ λ ) = 0. Expning ll terms we get equivlently (λ λ + λ λ λ λ λ λ ) (λ λ + λ λ λ λ λ λ ) +(λ λ + λ λ λ λ λ λ ) = 0. The lst eqution n e esily heke. Grssmnn-Plüker-Reltions n e interprete in mny equivlent wys n y this link severl rnhes o geometry n invrint theory. We will here present three more interprettions (or proos i you wnt).. Determinnt expnsion: The Grssmnn-Plüker-Reltion [, ][, ] [, ][, ] + [, ][, ] = 0 n e onsiere s eterminnt expnsion. For this ssume w.l.o.g. ( ) tht [, ] ( ) 0. Ater projetive trnsormtion we my ssume tht = 0 n = 0. The Grssmnn-Plüker-Reltion the res s 0 0 2 2 0 2 0 2 + 0 2 0 2 = 2 2 2 ( ) + 2 ( ) = 0 The lst expression is esily reognize s the expnsion ormul or the eterminnt n oviously evlutes to zero. 2. Are reltion: Ater ( ) projetive ( ) trnsormtion ( ) n resling ( ) we n lso ssume tht = 0, =, = n =. Then the Grssmnn-Plüker-Reltion res. 0 0 + 0 = ( ) ( ) + ( ) = 0 This ormul n e inely (!) interprete s the reltion o three irete length segments o 3 points,, on line
0 6 Determinnts ( ) ( ) ( ) 2. Crmer s rule: Let us ssume tht [, ] 0. Crmer s rule gives us n expliit ormul to solve the system o equtions ( ) ( ( ) α =. 2 2 β) 2 We get α = [, ] [, ] n β = [, ] [, ]. Inserting this into the originl eqution n multiplying y [, ] we get: [, ] +[, ] [, ] =0. Here 0 mens the zero vetor. Thus we n in the ollowing expnsion o zero. 0 = [[, ] +[, ] [, ], ] = [, ][, ] + [, ][, ] [, ][, ]. This is extly the Grssmnn-Plüker-Reltion. Wht hppens in imension 3 (i.e. the projetive plne)? First o ll we otin onsequene o the Grssmnn-Plüker-Reltion on the line when we the sme point to ny rket: Theorem 6.3. For ny vetors,,,, x R 3 the ollowing eqution hols: [x,, ][x,, ] [x,, ][x,, ] + [x,, ][x,, ] = 0 Proo. Assuming w.l.o.g tht x = (, 0, 0) reues ll eterminnts o the expression to 2 2 eterminnts ny o the ove proos trnsltes literlly. In the projetive plne we get nother Grssmnn-Plüker-Reltion tht involves our inste o three summns. Theorem 6.4. For ny vetors,,,, e, R 3 hols: the ollowing eqution [,, ][, e, ] [,, ][, e, ] + [,, e][,, ] [,, ][,, ] = 0. Proo. Applying Crmer s rule to the solution o 3 3 eqution e α β = γ
6.5 Grssmnn-Plüker Reltions e + e e = 0 Fig. 6.4. Grssmnn-Plüker-Reltion s re ormuls. we n prove the ientity: rerrnging the terms yiels [,, e] +[,, e] +[,, ] e =[,, e]. [, e, ] [, e, ] +[,, ] e [,, e] =0. Inserting this expnsion o the zero vetor 0 into [,, 0] = 0 yiels (ter expning the terms y multilinerity) the esire reltion. Agin, we n lso interpret this eqution in mny ierent wys. Setting (,, ) to the unit mtrix the Grssmnn-Plüker-Reltion enoes the evelopment o the 3 3 eterminnt (, e, ) y the irst olumn. We get: 0 0 0 0 0 0 e 2 e 2 2 3 e 3 0 0 2 3 0 0 0 e 0 e 2 2 3 e 3 + 0 e 0 e 2 3 0 0 e 0 0 2 2 3 3 0 0 2 3 0 0 0 e 0 2 e 2 3 3 e 3 e = 2 e 2 2 3 e 3 3 e e 2 + e 3 2 2 3 e 2 2 e = 0 2 3 Oserve tht we n express eh minor o the eterminnt [, e, ], s suitle rket tht involves,,. This point will lter on e o unmentl importne. There is lso nie interprettion tht generlizes the re-viewpoint. The eterminnt 2 2 2 lultes twie the oriente re (,, ) o the ine tringle,,. Ater suitle projetive trnsormtion the Grssmnn-Plüker-Reltion res s 0 e 0 2 2 e 2 2 0 0 0 e 0 2 2 e 2 2 0 0 + 0 e 0 e 2 2 2 2 0 0 0 e 0 2 2 2 e 2 0 0 = 0 In terms o tringle res this eqution res s (, e, ) (, e, )+ (,, ) (,, e) = 0 This ormul hs gin iret geometri interprettion in terms o ine oriente res o tringles. Assume tht,, e, re ny our points in the ine
2 6 Determinnts plne. The onvex hull o these our points n in two wys e overe y tringles spnne y three o the points. These two possiilities must oth result in the sme totl (oriente) re. This is the Grssmnn-Plüker-Reltion. Using Grssmnn-Plüker-Reltions we n esily explin why the property tht,, e meet oul e expresse either y or y Aing the two expressions yiels [,, e][,, ] [,, ][,, e] = 0 [,, ][e,, ] [,, ][e,, ] = 0 [,, ][e,, ] [,, ][e,, ] + [,, e][,, ] [,, ][,, e] Whih is extly Grssmnn-Plüker-Reltion. Hene this eqution must e zero, whih proves the equivlene o the two ove expressions.