Some remarks on a generalized vector product

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1 Revista Itegració Escuela de Matemáticas Uiversidad Idustrial de Satader Vol. 9, No., 011, pág Some remarks o a geeralized vector product Primitivo Acosta-Humáez a, Moisés Arada b, Reialdo Núñez c a Uiversidad del Norte, Departameto de Matemáticas, Barraquilla, Colombia. b Potificia Uiversidad Javeriaa, Departameto de Matemáticas, Bogotá, Colombia. c Uiversidad Sergio Arboleda, Escuela de Matemáticas, Bogotá, Colombia. Abstract. I this paper we use a geeralized vector product to costruct a exterior form :R k R k, where k! k!k!, k. Fially, for k 1 we itroduce the reversig operatio to study this geeralized vector product over palidromic ad atipalidromic vectors. Keywords: alteratig multiliear fuctio, atipalidromic vector, exterior product, palidromic vector, reversig, vector product. MSC000: 15A75, 15A7. Alguas observacioes sobre u producto vectorial geeralizado Resume. E este artículo usamos u producto vectorial geeralizado para costruir ua forma exterior :R k R k, e dode como es atural, k! k!k!, k. Fialmete, para k 1 itroducimos la operació reversar para estudiar este producto vectorial geeralizado sobre vectores palidrómicos y atipalidrómicos. Palabras claves: fució multilieal alterate, producto exterior, producto vectorial, reversar, vector palidrómico, vector atipalidrómico. Itroductio It is well kow that the vector product over R 3 is a alteratig -liear fuctio from R 3 R 3 oto R 3. Although this vector product is a atural topic to be studied i ay course of basic liear algebra, there is a plety of textbooks o this subject i where it is ot cosidered over R. The followig defiitio, with iterestig remarks, ca be foud also i [3, 7, 8]. Let A 1 a 11,a 1,...,a 1,..., A 1 a 11,a 1,...,a 1 0 Correspodig author: primi.acostahumaez@uiorte.edu.co. Received: July 30, 011, Accepted: August 30,

2 15 P. Acosta-Humáez, M. Arada & R. Núñez be 1 vectors i R. The vector product over R is a fuctio :R 1 R such that A 1,A,..., A 1 A 1 A A k det X k e k, 1 where e k is the k th uity vector of the stadard basis of R ad X k is the square matrix obtaied through the deletig of the k th colum of the a ij 1. Notice that i this case the fuctio is ot biary ad seds a matrix M of size 1 to a vector of its 1 maximal miors. Oe aim of this paper is to give a algorithm to costruct, usig elemetary techiques, a fuctio with domai i R k ad codomai R k which will be a alteratig k-liear fuctio that obviously geeralizes the previous vector product defied over R. Usig techiques ad methods of algebraic geometry we ca see that the vector product obtaied here, without sigs, correspods to the Plücker coordiates of the matrix M see [4, 5]. Although this vector product is kow ad useful to defie the cocept of Grassmaia variety see [4], we preset a alterative costructio, avoidig algebraic geometry, which lead us to kow results that ca be foud as for example i [6]. Aother aim of this work, followig [1, ], is the presetatio of some origial results cocerig the vector product for k 1 i palidromic ad atipalidromic vectors by meas of reversig operatio. The way this paper is preseted ca allow studets ad teachers of basic liear algebra the implemetatio of these results o their courses. This is our fial aim. 1. A geeralized vector product I this sectio we set some prelimiaries, properties ad the Cramer s rule as applicatio of the geeralized vector product. k Prelimiaries Followig [3, 7] we defie the geeralized vector product over R as the fuctio :R 1 R such that for A 1 a 11,a 1,...,a 1,...,A 1 a 11,a 1,...,a 1, 1 vectors of R, their vector product is give by A 1,A,..., A 1 A 1 A A k det X k e k, where e k is the k th elemet of the caoical basis for R ad X k is the square matrix obtaied after the elimiatio of the k th colum of the matrix a ij 1. The defiitio preseted i expressio correspods to a atural geeralizatio of the vector product of two vectors belogig to R 3. k1 [Revista Itegració

3 Some remarks o a geeralized vector product Some properties Let A 1,A,...,A be vectors of R. The followig statemets hold. 1 A 1, A,..., A 1 is a orthogoal vector for the give vectors. Assume α, β R, B i R ; the A 1 A αa i + βb i A 1 A 1 A αa i A 1 + A 1 A βb i A 1. 3 Let A be the matrix give by A A 1,A,...,A. The det A A 1 A A 1 1+j A j A 1 A j 1 A j+1 A. 4 The vectors A 1,A,...,A 1 are 1 liearly depedet vectors for R if ad oly if A 1 A A 1 0. It is well kow that these properties ca be prove usig the properties of the determiat see, for example, [3, 7] Cramer s rule Cosider the followig system of liear equatios that ca be expressed i vectorial way as a 11 x 1 + a 1 x + + a 1 x b 1 a 1 x 1 + a x + + a x b a 1 x 1 + a x + + a x b x 1 A 1 + x A + + x A B, 3 beig A i a 1i,a i,...,a i with i 1,,..., ad B b 1,b,...,b. Suppose that det A 1,A,...,A 0. Therefore the system has a uique solutio that ca be obtaied applyig the scalar product betwee the equatio 3 ad A A 3 A ; so we obtai x 1 A 1 + x A + + x A A A 3 A B A A 3 A, x 1 A 1 A A 3 A B A A 3 A,. sice A j A A 3 A 0for j, 3,...,. Therefore, x 1 B A A 3 A A 1 A A 3 A det B,A,A 3,,A deta 1,A,A 3,,A. 4 Vol. 9, No., 011]

4 154 P. Acosta-Humáez, M. Arada & R. Núñez I a geeral way, we ca obtai x i B A 1 A A i 1 A i+1 A A i A 1 A A i 1 A i+1 A that is, the well-kow Cramer s rule. 1i+1 det A 1,A,...,A i 1,B,A i+1,,a 1 i+1 deta 1,A,A 3,,A det A 1,A,...,A i 1,B,A i+1,,a, deta 1,A,A 3,,A. Didactic way to defie : algorithm ad properties I this sectio we propose a didactic way to defie the exterior product. To do this, we set a algorithm to the costructio of ad as cosequece of this costructio arise some properties..4. Algorithm to the costructio of Here we preset a algorithm ad some simple examples to illustrate it. Step 1 Cosider N ad 1 k, beig k a iteger. We defie I {i 1 i i k :1 i 1 <i < <i k } ; this meas that the elemets belogig to I are chais of umbers coformed i agreemet with the lexicographic order. Example.1. For 5ad k 3we have I {13, 14, 15, 134, 135, 145, 34, 35, 45, 345}. As we ca see, #I k Example.. For 5ad k, we obtai For istace, I is give by I {1, 13, 14, 15, 3, 4, 5, 34, 35, 45}. Step We set that I should be ordered lexicographically: I 1 <I < <I k. I this way, if I s I, the there exists p oly oe such that I s I p. Thus, we ca defie p as the rak of I s ad will be deoted by r I s p. That is, p is the place of I s i I as set of ordered elemets lexicographically. I Example.1 we ca see that r 34 7, r I Example. we have r 5 7, r [Revista Itegració

5 Some remarks o a geeralized vector product 155 Step 3 Let u 1 u 11,u 1,...,u 1,...,u k u k1,u k,...,u k, be k vectors of R, with k. Cosider the matrix U u ij of order k coformed by these vectors. Assume i 1 i i k I ad let U i1i i k be the matrix of order k, coformed by the k colums i 1,i,,i k of U. From ow o, U always will be a matrix of this kid. Example.3. Cosider U a 1 a a 3 a 4 a 5 b 1 b b 3 b 4 b 5 ; c 1 c c 3 c 4 c 5 i this case, U 13 a 1 a a 3 b 1 b b 3 ad U 45 a a 4 a 5 b b 4 b 5. c 1 c c 3 c c 4 c 5 Notice that whe we choose a particular umber of colums of such matrix U, which exactly correspods to delete i U the o-selected colums. Step 4 Cosider R k : R } R {{ R }. k times Now we defie the fuctio exterior product :R k R k as follows: U i I 1 k ri det U i e k ri+1, where e k ri+1 correspods to the k ri+1 th uity vector of the stadard basis of R k. For coveiece, we ca write U u 1,u,...,u k u 1 u... u k. Example.4. Cosider the vectors, 3, 1, 5, 4, 7,, 0 R 4. The vector, 3, 1, 5 4, 7,, 0 belogs to R 4 R 6. I this case I {1, 13, 14, 3, 4, 34}, U, so that U e e e e e e 1 e 6 +8e 5 + 0e e e 10e 1 10, 35, 13, 0, 8,. Vol. 9, No., 011]

6 156 P. Acosta-Humáez, M. Arada & R. Núñez Example.5. Cosider the caoical basis for R 4, that is, e 1 1, 0, 0, 0, e 0, 1, 0, 0, e 3 0, 0, 1, 0 ad e 1 0, 0, 0, 1. Thus, the exterior product e i e j for i<j is give by e 1 e 0, 0, 0, 0, 0, 1 e 6 R 6, e 1 e 3 0, 0, 0, 0, 1, 0 e 5 R 6, e 1 e 4 0, 0, 0, 1, 0, 0 e 4 R 6, e e 3 0, 0, 1, 0, 0, 0 e 3 R 6, e e 4 0, 1, 0, 0, 0, 0 e R 6, e 3 e 4 1, 0, 0, 0, 0, 0 e 1 R 6. As we ca see, the set B {e 1 e,e 1 e 3,e 1 e 4,e e 3,e e 4,e 3 e 4 } R 6 is a basis for R 6. Notice that i a give basis B for R, the exterior product of them take i sets of k-elemets without repetitio costitutes a basis B for R k..5. Some properties of The followig properties are satisfied by : 1 If k, the U det U. If k 1, the is the geeralized vector product. 3 If is eve ad k 1, the U is orthogoal to U. 4 is k liear: u 1,...,u i + b,...,u k u 1,...,u i,...,u k + u 1,...,b,...,u k. 5 If M p is a permutatio of two rows beig fixed the other oes of M, the M p M. 6 If u 1,...,u k are k liear depedet vectors of R, the u 1,...,u k 0 R k. Proof. We proceed accordig to each item. 1 Assumig k we have elemet. So, k 1 ad ri 1due to I has oly oe U 1 k ri det U i e k ri+1 i I detu i. Trivially we ca see that for R, e 1 1. [Revista Itegració

7 Some remarks o a geeralized vector product 157 Assumig k 1, we have ; i this way, I has elemets. k 1 Owig to the symmetry of, the electio of 1 colums of the matrix U k correspods to the elimiatio of oe colum of U precisely the avoided colum i the electio. I other words, we ca see that U i X ri+1, where X ri+1 correspods to the matrix that has bee obtaied throughout U deletig the ri+1-th colum, so that U i I 1 ri det U i e ri+1 1 ri+1+1 det U i e ri+1 i I 1 j+1 det X j e j j1 3 For p ad k 1, we have u 1... u k. p p; thus, the cardiality of I is eve ad 1 I {1,,...,p,p+1,...,p}. Furthermore, r i 1. I this way, U R p. O the other had, cosiderig U u 1,u,...,u p ad U v 1,v,...,v p, we obtai U 1 p i det U i e p i+1 i I p i1 1 i u i e p i+1 u p, u p 1,...,u, u 1, where it follows that v j 1 j+1 u p j+1 for j 1,,...,p. Therefore, U U u 1,u,...,u p 1,u p u p, u p 1,...,u, u 1 u 1 u p u u p u p 1 u u p u 1 u 1 u p u p u p+1 u p u p+1 u p+1 u p 0. Items 4, 5 ad 6 ca be prove usig the properties of the determiat i similar way as the previous oes. Vol. 9, No., 011]

8 158 P. Acosta-Humáez, M. Arada & R. Núñez 3. Reversig operatio over The reversig operatio has bee applied successfully over rigs ad vector spaces see [1, ]. I this sectio we apply the reversig operatio to obtai some results that ivolve the exterior product with the palidromic ad atipalidromic vectors. The followig results correspod to a geeralizatio of some results preseted i []. Cosider the matrix M m i,j of size m. The reversig of M, deoted by M, is give by M m i,j, where m i,j m i, j+1. We ca see that the size of M is m too. We deote by J I the reversig of the idetity matrix I of size. Thus, the followig properties ca be prove see []. 1. The double reversig: M mi,j m i, j+1 m i, j+1+1 mi,j M,. M MJ, 3. J J I. The followig defiitios were itroduced i []. A matrix M is called palidromic whether it satisfies M M. I the same way, a matrix M is called atipalidromic whether it satisfies M M. I particular, for m 1we get palidromic ad atipalidromic vectors, respectively. As we ca see, the palidromic matrix M satisfies m i,j m i, j+1, ad therefore M has at least pair of equal colums if is eve as well 1 whe is odd. This fact lead us to the followig result. { 1 /, k, k Z + Propositio 3.1. detj 1 +3, k 1, k Z +. Proof. We proceed by iductio over. Assumig 1, we have that I 1ad J 1, thus det J Let the propositio be true for ; thus we will prove that it is also true for +1. We start cosiderig that is eve, so we get det J det J Now, cosiderig as a positive odd iteger, we have det J det J [Revista Itegració

9 Some remarks o a geeralized vector product 159 Now, we study the relatioship betwee the exterior product ad the reversig operatio. We start cosiderig k 1, that is, the geeralized vector product over R. Cosider M 1 m 11,m 1,...,m 1,...,M 1 m 1,1,a 1,,...,m 1,, 1 vectors i R. The geeralized vector product is give by the equatio 1, therefore we obtai M 1,M,..., M k det M k e k ; 5 here e k is the k-th elemet of the caoical basis for R ad M k is the square matrix obtaied after the deletig of the k-th colum of the matrix M m ij 1. The matrix M k is a square matrix of size 1 1 ad is give by { M k m k mi,j, if j<k i,j m i,j+1, if j k. Propositio 3.. If we cosider M m ij 1, the M k M k+1 J 1, for 1 k. Proof. We kow that M MJ, that is, m i,j m i, j+1, 1 j. Therefore { mi,j M k m k, if j<k i,j mi,j+1, if j k k1 6 { mi, j+1, if j<k m i, j+1+1, if j k. O the other had, M k+1 m k+1 i,j { mi,j, if j< k +1 m i,j+1, if j k Now, we obtai M k+1 J 1 m k+1 i, 1 j+1 m k+1 i, j { mi, j, if j< k +1 m i, j+1, if j k +1 { mi, j, if j>k 1 m i, j+1, if j k 1 { mi, j, if j k m i, j+1, if j<k M k. The followig propositio is a geeralizatio of a result preseted i [], where it was aalyzed the reversig of the vector product i R 3. From ow o, for suitability we deote M M 1,M,...,M 1, i.e., M is the matrix that has as rows the vectors M 1,M,..., M 1 ; thus, we obtai M M 1, M,..., M 1. Vol. 9, No., 011]

10 160 P. Acosta-Humáez, M. Arada & R. Núñez I the same way, for suitability we write M M 1, M,..., M 1. Propositio 3.3. The geeralized vector product of M i, 1 i 1, satisfies 1 3 M 1, M,..., M 1, k, M M 1, M,..., M 1, k 1, where k Z +. Proof. For suitability we deote M M 1,M,...,M 1, i.e., M is the matrix that has as rows the vectors M 1,M,..., M 1 ; thus, we obtai I the same way, for suitability we write M M 1, M,..., M 1. M M 1, M,..., M 1. Now, applyig the geeralized vector product we obtai M 1 k+1 M k det e k k1 k1 k1 k1 det J 1 1 k+1 det M k+1 J 1 e k 1 k+1 det M k+1 J 1 e k 1 k+1 det M k+1 det J 1 e k k det J det J 1 1 k det M k e k+1 k1 k1 1 k+1 det M k e k+1 1 k+1 det M k e k J 1 +1 det J 1 M 1, M,..., M 1, [Revista Itegració

11 Some remarks o a geeralized vector product 161 ad therefore M M 1, M,..., M 1,k M 1, M,..., M 1, k 1 M 1,M,..., M 1, k M 1,M,..., M 1,k 1. If M is a palidromic matrix, the the miors M k have at least 1 pair of equal colums whe is eve, ad respectively 1 1 whe is odd. This implies that for 4, the miors have at least oe pair of equal colums ad therefore det M k 0 for all 1 k, so that M 1, M,..., M 1 0 R. 8 This meas that the geeralized vector product of 1 palidromic vectors i R is iterestig whe 1 3. The same result is obtaied whe we assume M as a atipalidromic matrix. Fial Remarks Whe we cosider the exterior product for k 1, the previous results caot be applied due to the fact that, i geeral, they are ot true. To illustrate it, we preset the followig example. Example 3.4. Cosider the vectors, 3, 1, 5 ad 4, 7,, 0 i R 4. I this case, M ad M As we have see before, Therefore,, 3, 1, 5 4, 7,, 0 10, 35, 13, 0, 8,. 5, 1, 3, 0,, 7, e e e e e e 1 10e e 5 0e e 3 4 4e e 1, 8, 13, 0, 35, 10. Vol. 9, No., 011]

12 16 P. Acosta-Humáez, M. Arada & R. Núñez Thus, i geeral, the exterior product does ot satisfies U 1 p U, for some p Z. Fially, although this paper is preseted i a didactic way, there are origial results correspodig to the relatios betwee the reversig operatio ad the geeralized vector product. Ackowledgemets The first author is partially supported by MICIIN/FEDER grat umber MTM ad by Uiversidad del Norte. The secod author is supported by Potificia Uiversidad Javeriaa. The third author is partially supported by Uiversidad Sergio Arboleda. The authors thaks to the aoymous referees by their useful commets ad suggestios. Refereces [1] Acosta-Humáez P., Chuque A. ad Rodríguez A., Pastig ad Reversig operatios over some rigs, Boletí de Matemáticas, , [] Acosta-Humáez P., Chuque A. ad Rodríguez A., Pastig ad Reversig operatios over some vector spaces, Preprit 011, 3p. [3] Arada M. ad Núñez R., The Cramer s rule via geeralized vector product over R Spaish, Uiversitas Scietorum, Ivestigacioes Matemáticas, 8 003, [4] Harris J., Algebraic Geometry, A First Course, Spriger, New York, 199. [5] Hodge W.V.D. ad Pedoe D., Methods of Algebraic Geometry, I, Cambridge Uiversity Press, [6] Lag S., Liear Algebra, Udergraduate Texts i Mathematics, Spriger, New York, [7] Marmolejo M., Vector product over R : The Lagrage s geeral idetity Spaish, Matemáticas eseñaza uiversitaria, , [8] Olivert J., Structures of multiliear algebra Spaish, Uiversidad de Valecia, Valecia, [Revista Itegració