Skew-symmetric matrices related to the vector cross product in c7

DOI: 10.2478/auom-2023-0003 An. S¸ . Uni . O idius Cons an ¸a Vol. 31(1),2023,47–69 Skew-symme ic ma ices ela ed o he ec o c oss p oduc in C7 P. D. Bei es, A. P. Nicol´as, Jos´e Vi ´o ia Abs ac Skew-symme ic ma ices o o de 7 de ined h ough he 2- old ec o c oss p oduc in C7, and o he ela ed ma ices, a e p esen ed. Mo e conc e ely, ma ix p ope ies, namely in e ibili y, nullspace, powe s and index, a e s udied. As a consequence, esul s on ec o c oss p oduc equa ions, ec o c oss p oduc di e en ial equa ions and ec o c oss p oduc di e ence equa ions in C7a e es ablished. 1 In oduc ion Assuming he usual de ini ion, as explained by Elduque in he elemen a y ac- coun [13] on ec o c oss p oduc s and hei connec ions wi h he excep ional basic classical simple Lie supe algeb as, - old ec o c oss p oduc s exis only o d-dimensional ec o spaces wi h: = 1 and de en; = 2 and d= 3 o 7; = 3 and d= 8; and =d−1 o an a bi a y d. The i s p oo o his classical esul , and an ex ension o i , goes back o he wo k [7], whe e B own and G ay p esen ed an algeb aic p oo . An algeb aic- opologic p oo o he same esul o eal euclidean spaces was gi en by Eckmann, who in [12] assumed con inui y – a weake condi ion – ins ead o mul ilinea i y. Based on he esul s in [12], a a ia ion o he la e p oo was gi en by Whi ehead in [27]. In addi ion, in [16], ci ing he a icles [12] and [27], G ay es ablished Key Wo ds: 2- old ec o c oss p oduc , He mi ian inne p oduc , Skew-symme ic ma ix, Gene alized in e se, (Vec o c oss p oduc , Vec o c oss p oduc di e en ial, Vec o c oss p oduc di e ence) equa ion 2010 Ma hema ics Subjec Classi ica ion: P ima y 15A72; Seconda y 15B57, 15A09. Recei ed: 05.04.2022 Accep ed: 15.07.2022 47 SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C748 esul s abou ec o c oss p oduc s on mani olds. An elemen a y p oo o he classical esul , al hough only alid o e a ield o cha ac e is ic 0, was gi en by Ros in [24]. La e on, Meybe g simpli ied his p oo in [23]. The men ioned classical esul can be seen as a consequence o ano he clas- sical esul on he classi ica ion o Hu wi z algeb as ( ha is, uni al composi ion algeb as, [2], [15], [18]). The eal and complex cases a e due o Hu wi z, who p esen ed he classi ica ion in [19]. Jacobson es ablished he classi ica ion, in [21], o e a ield Fo cha ac e is ic di e en om 2. Mo e conc e ely, he gene alized Hu wi z Theo em asse s ha , o e F, i Ais a ini e dimensional composi ion algeb a wi h iden i y, hen i s dimension is equal o 1, 2, 4 o 8. Fu he mo e, as Jacobson was in e es ed in he s udy o he au omo phisms o Hu wi z algeb as, he p o ed ha Ais isomo phic ei he o he base ield, a sepa able quad a ic ex ension o he base ield (a quad a ic commu a i e and associa i e sepa able algeb a), a gene alized qua e nion algeb a (a ou - dimensional algeb a ha is associa i e bu no commu a i e) o a gene alized oc onion algeb a (also called Cayley algeb a: an eigh -dimensional algeb a ha is al e na i e bu no associa i e), [21]. Th oughou he yea s, he in e es in 2- old ec o c oss p oduc s has e- mained ali e. In [20], Ik amo s udies he complex ec o c oss p oduc in C3. Cos a, Facas Vicen e, Bei es, Ma ins, Se ˆodio and Tadeu, in [10], use he ec o c oss p oduc in R7 o s udy he o hogonal p ojec ion o a poin on o a line. In [9], Ca a ino and Vi ´o ia exp ess he dis ance be ween wo skew lines in R7in e ms o he double ec o c oss p oduc . In [3], ec o c oss p oduc di e en ial and di e ence equa ions a e s udied by Bei es, Nicol´as, Sa ai a and Vi ´o ia. A gene aliza ion o he s anda d de ini ion o 2- old ec- o c oss p oduc is p oposed in [22] by Lewin an. In [5], Bei es, Nicol´as and Vi ´o ia pu sue an a i hme ic o closed balls in Rnwhich includes ope a ions in ol ing he 2- old ec o c oss p oduc . Using his p oduc in R3, Bei es and Ca a ino es ablish Gelin-Ces`a o’s iden i y o Leona do qua e nions in [1]. Fe ei a, Kaygo odo and Kudaybe geno desc ibe de i a ions o complex Filippo algeb as whose ealiza ions gene alize he 3-dimensional 2- old ec o c oss p oduc , [14]. The s uc u e o he p esen wo k, di ided in o h ee main sec ions, is as ollows. In sec ion 2, whe e some backg ound is p esen ed, known de i- ni ions, esul s and no a ions ela ed o he 2- old ec o c oss p oduc , o he 7-dimensional complex ec o space C7, o gene alized in e ses and o di e en ial and di e ence equa ions a e ecalled. In sec ion 3, p ope ies o ma ices ela ed o he 2- old ec o c oss p oduc in C7, namely on in e - ibili y, nullspace, powe s and index, a e es ablished. Pa ially ollowing he ideas o Agudo o R3in [11], whe e he uses he e m “ ec o di ision”, ec o c oss p oduc equa ions in C7a e conside ed in sec ion 4. Mo eo e , in C7, SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C749 ec o c oss p oduc di e en ial equa ions and ec o c oss p oduc di e ence equa ions a e s udied. Se e al esul s p esen ed in he wo ks [3] – o Bei es, Nicol´as, Sa ai a and Vi ´o ia –, [4] – due o Bei es, Nicol´as and Vi ´o ia –, [17] – whose au ho s a e G oss, T enkle and T oschke –, [25] – o T enkle –, and [26] – by T enkle and T enkle – a e ex ended. 2 P elimina ies Le Vbe a d-dimensional ec o space o e a ield Fo cha ac e is ic di e - en om 2, endowed wi h a nondegene a e symme ic bilinea o m (·,·). A bilinea map ×:V2→Vis a 2- old ec o c oss p oduc in Vi , o any u, ∈V: 1. (u× , u) = (u× , ) = 0, 2. (u× , u × ) =  (u, u) (u, ) ( , u) ( , ) . Recall ha 1. implies he skew-symme y o he ilinea map (· × ·,·), and so he an icommu a i i y o ×, [13]. In he p esen a icle, he 2- old ec o c oss p oduc in he 7-dimensional complex ec o space C7, deno ed by ×, is conside ed. Equip he 7-dimensional complex ec o space C7wi h he s anda d He - mi ian inne p oduc h·,·i :C7×C7→Cde ined by hx, yi= 7 X =1 x y , o all x=x1. . . x7T,y=y1. . . y7T∈C7. I sa is ies, espec- i ely, linea i y in he i s coo dina e, He mi ian (o conjuga e) symme y and posi i e de ini eness: hαx +βy, zi=αhx, zi+βhy, zi,(1) hx, yi=hy, xi,(2) hx, xi ≥ 0 and hx, xi= 0 ⇔x= 0.(3) Recall ha (2) implies ha hx, xi ∈ R. Recall also ha (1) and (2) imply conjuga e linea i y in he second coo dina e, ha is, hx, αy +βzi=αhx, yi+βhx, zi.(4) SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C750 When conside ing he 2- old ec o c oss p oduc in he 7-dimensional complex ec o space C7, obse e ha he nondegene a e symme ic bilinea o m (·,·) e e ed in he i s de ini ion is de ined by (x, y) = hx, yi, o all x=x1. . . x7T,y=y1. . . y7T∈C7. Th oughou he wo k, Cm×ndeno es he se o all m×ncomplex ma ices. When n= 1, Cm×1is iden i ied wi h Cm. When m=n= 1, C1×1is iden i ied wi h C. Le B∈Cm×n. A ma ix B(1) ∈Cn×mis a gene alized in e se o Bi BB(1)B=B. See [6] o mo e de ails on gene alized in e ses, also known as (1)-in e ses o g-in e ses, whe e he subsequen esul appea s. Theo em 1 ([6]).Le B∈Cm×n,b∈Cm. Then, he equa ion Bx =bis consis en i and only i , o some B(1),BB(1)b=b. Le A∈Cn×n. The index Ind(A) o Ais he smalles l∈N0such ha R(Al) = R(Al+1) o , equi alen ly, N(Al) = N(Al+1), whe e Rand Ns and o he column space (o ange) and he nullspace, [8]. Al e na i ely, bu equi alen ly, i can also be de ined as he smalles l∈N0such ha Cn=R(Al)⊕N(Al). Le Ind(A) = l. The D azin in e se o Ais he unique ma ix AD∈Cn×n which sa is ies AAD=ADA, ADAAD=AD, Al+1AD=Al. When Ind(A)∈ {0,1},ADis some imes called he g oup-in e se o Aand he las equali y assumes he o m AADA=A. The e a e se e al me hods o compu ing AD, as desc ibed in [8] and e e ences he ein, some o which equi e all eigen alues o be well de e mined. Le A, B ∈Cn×nand 0∈R. Le = ( ) be a Cn- alued unc ion o he eal a iable . Th oughou he wo k, x=x( ) s ands o an unknown Cn- alued unc ion o he eal a iable and ˙x=dx d deno es he co esponding de i a i e ec o o x. A ec o x0∈Cnis a consis en ini ial ec o o he di e en ial equa ion A˙x+Bx = (5) i he ini ial alue p oblem A˙x+Bx = , x( 0) = x0,(6) possesses a leas one solu ion. In his case, x( 0) = x0is said o be a consis en ini ial condi ion. Fu he , (5) is called ac able i (6) has a unique solu ion o each consis en ini ial ec o x0, [8]. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C751 Theo em 2. [8] Le A, B ∈Cn×n. The homogeneous di e en ial equa ion A˙x+Bx = 0 is ac able i and only i (λA +B)−1exis s o some λ∈C. Le A, B ∈Cn×n. Le (k)= (k)( )∈Cnbe he k- h e m o a sequence o ec o s, k= 0,1,2, .... Th oughou he wo k, x(k)=x(k)( )∈Cns ands o he k- h e m o an unknown sequence o ec o s, k= 0,1,2, . . . We assume ha x(0) =x0is gi en. A ec o x0∈Cnis a consis en ini ial ec o o he di e ence equa ion Ax(k+1) =Bx(k)+ (k)(7) i he ini ial alue p oblem Ax(k+1) =Bx(k)+ (k), k = 1,2, . . . , x(0) =x0,(8) has a solu ion o x(k). In his case, x(0) =x0is said o be a consis en ini ial condi ion. Fu he mo e, (7) is called ac able i (8) has a unique solu ion o each consis en ini ial ec o x0, [8]. Theo em 3. [8] Le A, B ∈Cn×n. The homogeneous di e ence equa ion Ax(k+1) =Bx(k)is ac able i and only i (λA +B)−1exis s o some λ∈C. 3 P ope ies Le a=a1a2a3a4a5a6a7T∈C7. Conside he linea map- ping a×:C7→C7 x7→ a×(x) = a×x. Fo each a∈C7, he e exis s a unique ma ix Sa∈C7×7such ha a×x=Sax, (9) whe e Sa=           0−a3a2−a5a4−a7a6 a30−a1−a6a7a4−a5 −a2a10a7a6−a5−a4 a5a6−a70−a1−a2a3 −a4−a7−a6a10a3a2 a7−a4a5a2−a30−a1 −a6a5a4−a3−a2a10           .(10) In he ollowing esul , some p ope ies ela ed o he ma ices de ined in (9)-(10) a e es ablished. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C752 P oposi ion 4. Le a, b, c ∈C7. Le α, β ∈C. Then: 1. Sαa+βbc=αSac+βSbc; 2. Sa=Sa; 3. Sa=−ST a; 4. S∗ a=−Sa, whe e ·∗s ands o he conjuga e anspose o a ma ix; 5. Sab=−Sba; 6. Saa= 0; 7. Saa= 2i           Im(a2a3) + Im(a4a5) + Im(a6a7) −Im(a1a3) + Im(a4a6)−Im(a5a7) Im(a1a2)−Im(a4a7)−Im(a5a6) −Im(a1a5)−Im(a2a6) + Im(a3a7) Im(a1a4) + Im(a2a7) + Im(a3a6) −Im(a1a7) + Im(a2a4)−Im(a3a5) Im(a1a6)−Im(a2a5)−Im(a3a4)           ; 8. Sab=Sab; 9. Sais singula ; 10. S2 a=aaT− ha, aiI7; 11. S3 a=−ha, aiSa; 12. he eigen alues o Saa e 0,p|ha, ai|eiθ 2and p|ha, ai|ei(θ 2+π), wi h θan a gumen o −ha, ai; 13. he nullspace o Sa, whe e a6= 0, is N(Sa) = {αa :α∈C}. P oo . P ope ies 1. and 5. a e di ec consequences o , espec i ely, he bilin- ea i y and he an icommu a i i y o ×in C7. F om (10) i is s aigh o wa d o p o e 2. and 3. Conce ning 4., in oking 2. and 3. leads o S∗ a= (Sa)T= (Sa)T=−Sa. Taking b=ain 5. leads o 6. By p ope y 5., Saa+Saa= 0 which, by 2., is equi alen o Saa+Saa= 0⇔Saa+Saa= 0. The las equali y means ha each en y o Saais ei he ze o o a pu ely imagina y complex numbe . Conc e ely, om (10), Saais he ma ix SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C753           a2a3−a3a2+a4a5−a5a4+a6a7−a7a6 −a1a3+a3a1+a4a6−a6a4−a5a7+a7a5 a1a2−a2a1−a4a7+a7a4−a5a6+a6a5 −a1a5+a5a1−a2a6+a6a2+a3a7−a7a3 a1a4−a4a1+a2a7−a7a2+a3a6−a6a3 −a1a7+a7a1+a2a4−a4a2−a3a5+a5a3 a1a6−a6a1−a2a5+a5a2−a3a4+a4a3           =           2iIm(a2a3)+2iIm(a4a5)+2iIm(a6a7) −2iIm(a1a3)+2iIm(a4a6)−2iIm(a5a7) 2iIm(a1a2)−2iIm(a4a7)−2iIm(a5a6) −2iIm(a1a5)−2iIm(a2a6)+2iIm(a3a7) 2iIm(a1a4)+2iIm(a2a7)+2iIm(a3a6) −2iIm(a1a7)+2iIm(a2a4)−2iIm(a3a5) 2iIm(a1a6)−2iIm(a2a5)−2iIm(a3a4)           , om whe e 7. ollows. Applying 2. allows o a i e a 8. since Sab=Sab. As a as 9., on he one hand, i a= 0 hen Sa= 0, a singula ma ix. On he o he hand, i a6= 0 hen, om 6., Saa= 0. I Sawe e in e ible hen a= 0, a con adic ion. As S2 a= [sij]7×7wi h sij =       − 7 X =1, 6=i a2 i i=j aiaji i6=j , aaT= [dij]7×7wi h dij =a2 ii i=j aiaji i6=j, and ha, ai= 7 X =1 a2 , hen 10. ollows. Taking in o accoun 10., S3 a=aaTSa− ha, aiSa. By 3. and 6., aaTSa= a(ST aa)T=−a(Saa)T= 0. Hence, 11. ollows. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C754 Rega ding 12., he cha ac e is ic equa ion o Sais de (Sa−λI7)=0 ⇔ −λ(λ2+ha, ai)3= 0 ⇔λ= 0 ∨λ2=−ha, ai ⇔λ= 0 ∨λ=p|ha, ai|eiθ 2∨λ=p|ha, ai|ei(θ 2+π), wi h θan a gumen o −ha, ai. Le a∈C7 {0}. The inclusion ⊇in 13. ollows om p ope y 6. since, o all γ∈C,Sa(γa) = γSaa= 0. By he p oo o 12., he eigen alue 0 has algeb aic mul iplici y 1. As 0 6=a∈N(Sa), he geome ic mul iplici y o 0 is 1. Hence, dim N(Sa) = dim {αa :α∈C}= 1, and 13. is ob ained. The subsequen esul s conce n powe s and aces o he ma ices de ined in (9)-(10). Lemma 5. Le a∈C7such ha ha, ai 6= 0. Fo m∈N, S2m a= (−1)m+1ha, aim−1aaT+ (−1)mha, aimI7(11) and S2m+1 a= (−1)mha, aimSa.(12) P oo . The p oo goes by induc ion on m. Fo (11), by 10. in P oposi ion 4, he base case holds. Also om 10. in P oposi ion 4 and he induc ion hypo hesis, we ha e S2(m+1) a=S2m aS2 a = [(−1)m+1ha, aim−1aaT+ (−1)mha, aimI7](aaT− ha, aiI7) = (−1)m+1ha, aimaaT−(−1)m+1ha, aimaaT +(−1)mha, aimaaT−(−1)mha, aim+1I7 = (−1)m+2ha, aimaaT+ (−1)m+1ha, aim+1I7, and he induc ion s ep holds oo. Fo (12), by 11. in P oposi ion 4, i is s aigh o wa d o see ha he base case holds. As o he induc ion s ep, by 10. in P oposi ion 4 and he induc ion hypo hesis, we ob ain S2m+3 a=S2m+1 aS2 a = (−1)mha, aimSa(aaT− ha, aiI7) = (−1)mha, aim(Saa)aT+ (−1)m+1ha, aim+1Sa. F om he e, aking in o accoun 6. in P oposi ion 4, he second pa o he esul ollows. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C755 Theo em 6. Le a∈C7such ha ha, ai 6= 0. Fo m∈N, (S2m+1 a) = 0 and (S2m a) = 6(−1)mha, aim.(13) P oo . F om (12) in Lemma 5, i is clea ha (S2m+1 a)=(−1)mha, aim (Sa) = 0. F om (11) in Lemma 5, aking in o accoun aaTw i en o he p oo o 10. in P oposi ion 4, (S2m a)=(−1)m+1ha, aim−1 (aaT)+(−1)mha, aim (I7) =−(−1)mha, aim+ 7(−1)mha, aim, and he exp ession o he ace o S2m ain (13) is ob ained. The ollowing esul s a e de o ed o gene alized in e ses, in e ibili y and in e ses o ma ices ela ed o he ma ices de ined in (9)-(10). Theo em 7. Le a∈C7such ha ha, ai 6= 0. A gene alized in e se o Sais S(1) a=−ha, ai−1Sa.(14) P oo . Wi h ha, ai 6= 0, 11. in P oposi ion 4 leads o (14) since Sa−ha, ai−1SaSa=−ha, ai−1S3 a=Sa. P oposi ion 8. Le a, b ∈C7and γ∈C. The ma ix γSa+Sbis singula . P oo . As Saand Sba e skew-symme ic ma ices, hen, o any γ∈C,γSa+ Sbis also skew-symme ic o odd o de . Hence, de (γSa+Sb) = 0. Lemma 9. Le a∈C7and α∈C. The ma ix Sa+αI7is non-singula i and only i α6= 0 and αis no a squa e oo o −ha, ai. P oo . A s aigh o wa d calcula ion o de (Sa+αI7) leads o α(α2+ha, ai)3. In he s a ed condi ions, de (Sa+αI7) = 0 i and only i α= 0 o α2= −ha, ai. Theo em 10. Le a∈C7. Le α∈C {0}such ha αis no a squa e oo o −ha, ai. Then (Sa+αI7)−1=−(α2+ha, ai)−1(Sa−αI7−α−1aaT).(15) SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C762 independen o he used λ. Hence, in wha ollows, we d op he subsc ip s λ and ake λ= 0. F om Theo em 12, Ind( ˆ Sa) = 1. In oking [8, Theo em 9.2.3, p. 175], we ob ain he necessa y and su icien condi ion x0∈R(ˆ Sa) = R(ˆ SD aˆ Sa) o a ec o x0∈C7 o be a consis en ini ial ec o o (27). Since ˆ SD aˆ Sa=ˆ Saˆ SD a, we ge (28). As ˆ Sa=B−1Sa, hen, by (15) in Theo em 10, we ob ain (29). Assume now ha x0∈C7is a consis en ini ial ec o o (27). As ˆ B=I7, once again om [8, Theo em 9.2.3], he unique solu ion o he homogeneous ini ial alue p oblem Sa˙x+Bx = 0, x( 0) = x0, is gi en by (30). Theo em 27. Le a∈C7wi h ha, ai 6= 0,b∈C7 {0}and α∈C {0}such ha αis no a squa e oo o −hb, bi. Le = ( )be a C7- alued unc ion o he eal a iable , con inuously di e en iable a ound 0, and le x=x( ) an unknown C7- alued unc ion o he eal a iable . A ec o x0∈C7is a consis en ini ial ec o o he ec o c oss p oduc di e en ial equa ion a×˙x+b×x+αx = (31) i and only i x0is o he o m x0= (I−ˆ Saˆ SD a)ˆ ( 0) + ˆ Saˆ SD aq, (32) o some ec o q∈C7, whe e ˆ Sa=−(α2+hb, bi)−1Sb−αI7−α−1bbTSa(33) and ˆ =−(α2+hb, bi)−1Sb−αI7−α−1bbT . (34) Mo eo e , i x0∈C7is a consis en ini ial ec o o (31), hen he unique solu ion o (31), wi h ini ial condi ion x( 0) = x0, is x( ) = e−ˆ SD a( − 0)ˆ Saˆ SD ax0+e−ˆ SD a Z 0 eˆ SD asˆ SD aˆ (s)ds+(I7−ˆ Saˆ SD a)ˆ ( ).(35) P oo . By (9), we can ew i e equa ion (31) as Sa˙x+ (Sb+αI7)x= , whe e α∈C {0}is such ha α26=−hb, bi. As in he p oo o Theo em 26, le B=Sb+αI7,ˆ Sa=B−1Sa,ˆ B=I7,ˆ =B−1 . Taking in o accoun Theo em 12, Ind( ˆ Sa) = 1. The necessa y and su icien condi ion x0∈ {(I7−ˆ Saˆ SD a)ˆ ( 0) + R(ˆ SD aˆ Sa)} o a ec o x0∈C7 o be a consis en ini ial ec o o (31) comes om [8, Theo em 9.2.3, p. 175], which leads o (32). By (15) in Theo em 10, we ob ain (33) and (34). Suppose now ha x0∈C7is a consis en ini ial ec o o (31). Once again om [8, Theo em 9.2.3], he unique solu ion o he inhomogeneous ini ial alue p oblem Sa˙x+Bx = , x( 0) = x0, is gi en by (35). SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C763 4.3 Vec o C oss P oduc Di e ence Equa ions In he p esen sec ion, some ec o c oss p oduc di e ence equa ions in C7 a e s udied. Theo em 28. Le b∈C7such ha hb, bi 6= 0 and le x(k)∈C7be he k- h e m o an unknown sequence o ec o s, k= 0,1,2, ... The unique solu ion o he ec o c oss p oduc di e ence equa ion x(k+1) =b×x(k),(36) wi h ini ial condi ion x(0) =x0, is x(k)=     x0, k = 0 (−1)k−1 2βk−1Sbx0, k ∈N, odd (−1)k 2+1βk−2bbT+ (−1)k 2βkI7x0, k ∈N, e en (37) whe e β=|hb, bi|1/2eiθ 2, wi h θan a gumen o hb, bi. P oo . Due o (9), equa ion (36) assumes he o m x(k+1) =Sbx(k), which is a ac able equa ion by Theo em 3. In ac , om Lemma 9, (λI7+Sb)−1exis s o e e y λ∈C {0} ha is no a squa e oo o −hb, bi. Taking in o accoun he ecu ence ela ion, he unique solu ion o he homogeneous ini ial alue p oblem x(k+1) =Sbx(k),k= 0,1,2, . . . ,x(0) =x0, is gi en by x(k)=Sk bx0, k = 0,1,2, ... F om Lemma 5, we a i e a (37). Theo em 29. Le b∈C7such ha hb, bi 6= 0. Le (k)∈C7be he k- h e m o a sequence o ec o s, k= 0,1,2, ..., and le x(k)∈C7be he k- h e m o an unknown sequence o ec o s, k= 0,1,2, .... The unique solu ion o he ec o c oss p oduc di e ence equa ion x(k+1) =b×x(k)+ (k),(38) wi h ini ial condi ion x(0) =x0, is x(k)=                x0, k = 0 (−1)k−1 2βk−1Sbx0+ k−1 X i=0 Sk−1−i b (i), k ∈N, odd (−1)k 2+1βk−2bbT+ (−1)k 2βkI7x0+ k−1 X i=0 Sk−1−i b (i), k ∈N, e en (39) whe e β=|hb, bi|1/2eiθ 2, wi h θan a gumen o hb, bi. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C764 P oo . Again by (9), equa ion (38) assumes he o m x(k+1) =Sbx(k)+ (k). The ecu ence ela ion allows o ob ain he unique solu ion o he inhomoge- neous ini ial alue p oblem x(k+1) =Sbx(k)+ (k),k= 0,1,2, . . . ,x(0) =x0, gi en by x(k)=Sk bx0+ k−1 X i=0 Sk−1−i b (i), k = 1,2, ... (40) F om Lemma 5, we ob ain (39). Co olla y 30. Le b∈C7such ha hb, bi 6= 0,c∈C7and le x(k)∈C7be he k- h e m o an unknown sequence o ec o s, k= 0,1,2, .... The unique solu ion o he ec o c oss p oduc di e ence equa ion x(k+1) =b×x(k)+c, (41) wi h ini ial condi ion x(0) =x0, is x(k)=                x0, k = 0 (−1)k−1 2βk−1Sbx0+ k−1 X i=0 Si bc, k ∈N, odd (−1)k 2+1βk−2bbT+ (−1)k 2βkI7x0+ k−1 X i=0 Si bc, k ∈N, e en (42) whe e β=|hb, bi|1/2eiθ 2, wi h θan a gumen o hb, bi. P oo . A pa icula case o he p e ious esul , pu ing cins ead o he se- quence  (k)k∈N0. Theo em 31. Le a, b ∈C7 {0}and le x(k)∈C7be he k- h e m o an unknown sequence o ec o s, k= 0,1,2, ... The ec o c oss p oduc di e ence equa ion a×x(k+1) =b×x(k)(43) is no ac able. P oo . F om (9), he ew i ing o equa ion (43) leads o Sax(k+1) =Sbx(k). F om P oposi ion 8, o any λ∈C,λSa+Sbis a singula ma ix and he esul ollows om Theo em 3. Simila ly o subsec ion 4.2, due o he p e ious esul , pe u bed e sions o he di e ence equa ion (43) a e now s udied. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C765 Theo em 32. Le a∈C7wi h ha, ai 6= 0,b∈C7 {0}and α∈C {0}such ha αis no a squa e oo o −hb, bi. Le x(k)∈C7be he k- h e m o an unknown sequence o ec o s, k= 0,1,2, .... A ec o x0∈C7is a consis en ini ial ec o o he ec o c oss p oduc di e ence equa ion a×x(k+1) =b×x(k)+αx(k)(44) i and only i x0is o he o m x0=ˆ Saˆ SD aq, (45) o some q∈C7, whe e ˆ Sa=−(α2+hb, bi)−1Sb−αI7−α−1bbTSa.(46) Mo eo e , i x0∈C7is a consis en ini ial ec o o (44), hen he unique solu ion o (44), wi h ini ial condi ion x(0) =x0, is x(k)=ˆ SD ak x0, k = 0,1,2, . . . (47) P oo . F om (9), equa ion (44) assumes he o m Sax(k+1) =Bx(k)whe e B=Sb+αI7wi h α∈C {0}such ha αis no a squa e oo o −hb, bi. By Lemma 9, Bis non-singula . Owed o his ac , λSa+Bis also a non-singula ma ix i λ= 0 and, by Theo em 3, (44) is a ac able equa ion. Following he no a ion in [8], le ˆ Sa,λ = (λSa+B)−1Saand ˆ Bλ= (λSa+B)−1B, whe e λ∈Cis such ha λSa+Bis non-singula . By [8, Theo em 9.2.2, p. 174], he consis ency o an ini ial ec o o (44) and i s gene al solu ion a e independen o he used λ. Hence, in wha ollows, we d op he subsc ip s λ and ake λ= 0. By Theo em 12, Ind( ˆ Sa) = 1. In oking [8, Theo em 9.3.2, p. 182-183], we ge he necessa y and su icien condi ion x0∈R(ˆ Sa) = R(ˆ SD aˆ Sa) o a ec o x0∈C7 o be a consis en ini ial ec o o (44). As ˆ SD aˆ Sa=ˆ Saˆ SD a, we ob ain (45). Since ˆ Sa=B−1Sa, hen, by (15) o Theo em 10, we a i e a (46). Suppose now ha x0∈C7is a consis en ini ial ec o o (44). Since ˆ B= I7, once again om [8, Theo em 9.3.2], he unique solu ion o he homogeneous ini ial alue p oblem Sax(k+1) =Bx(k),k= 0,1, . . . ,x(0) =x0, is gi en by (47). Theo em 33. Le a∈C7wi h ha, ai 6= 0,b∈C7 {0}and α∈C {0}such ha αis no a squa e oo o −hb, bi. Le (k)∈C7be he k- h e m o SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C766 a sequence o ec o s, k= 0,1,2, ..., and le x(k)∈C7 he k- h e m o an unknown sequence o ec o s, k= 0,1,2, .... A ec o x0∈C7is a consis en ini ial ec o o he ec o c oss p oduc di e ence equa ion a×x(k+1) =b×x(k)+αx(k)+ (k), k = 0,1,2,..., (48) i and only i x0is o he o m x0=−I7−ˆ Saˆ SD aˆ (0) +ˆ Saˆ SD aq, (49) o some q∈C7, whe e ˆ Sa=−(α2+hb, bi)−1Sb−αI7−α−1bbTSa(50) and ˆ (k)=−(α2+hb, bi)−1Sb−αI7−α−1bbT (k).(51) Mo eo e , i x0∈C7is a consis en ini ial ec o o (48), hen he unique solu ion o (48), wi h ini ial condi ion x(0) =x0, is x(k)gi en by      x0, k = 0 ˆ SD akˆ Saˆ SD ax0+ˆ SD a k−1 X i=0 ˆ SD ak−i−1ˆ (i)−I7−ˆ Saˆ SD aˆ (k), k = 1,2, . . . (52) P oo . By (9), he ew i ing o equa ion (48) leads o Sax(k+1) =Bx(k)+ (k), whe e B=Sb+αI7wi h α∈C {0}such ha α26=−hb, bi. As in he p oo o Theo em 32, le ˆ Sa=B−1Sa,ˆ B=I7,ˆ (k)=B−1 (k). F om Theo em 12, Ind( ˆ Sa) = 1. The necessa y and su icien condi ion x0∈ {−(I7−ˆ Saˆ SD a)ˆ (0) +R(ˆ SD aˆ Sa)} o a ec o x0∈C7 o be a consis en ini ial ec o o (48) comes om [8, Theo em 9.3.2, p. 182-183]. Thus, we ob ain (49). By (15), we ge (50) and (51). Assume now ha x0∈C7is a consis en ini ial ec o o (48). Once again om [8, Theo em 9.3.2], he unique solu ion o he inhomogeneous ini ial alue p oblem Sax(k+1) =Bx(k)+ (k),k= 0,1,2, . . . ,x(0) =x0, is gi en by (52). Acknowledgmen P. D. Bei es was suppo ed by FCT (Funda¸c˜ao pa a a Ciˆencia e a Tecnolo- gia, Po ugal), esea ch p ojec UIDB/00212/2020 o CMA-UBI (Cen o de Ma em´a ica e Aplica¸c˜oes, Uni e sidade da Bei a In e io , Po ugal), and by he esea ch p ojec MTM2017-83506-C2-2-P, Spain. The au ho A. P. Nicol´as was suppo ed by he la e esea ch p ojec . SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C767 Re e ences [1] P. D. Bei es, P. Ca a ino, On he Leona do qua e nion sequence. (2021), submi ed. [2] P. D. Bei es, A. P. Nicol´as, A no e on s anda d composi ion algeb as o ypes II and III. Ad ances in Applied Cli o d Algeb as 27 (2017), 955–964. [3] P. D. Bei es, A. P. Nicol´as, P. Sa ai a, J. 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Bei es Depa amen o de Ma em´a ica and CMA-UBI Uni e sidade da Bei a In e io R. Ma quˆes d’´ A ila e Bolama 6201-001 Co ilh˜a, Po ugal ORCID iD: h ps://o cid.o g/0000-0003-0266-7055 Email: pb[email p o ec ed] A. P. Nicol´as Depa amen o de Ma em´a icas Uni e sidad de O iedo Calle Fede ico Ga c´ıa Lo ca, 18 33007 O iedo, Espa˜na ORCID iD: h ps://o cid.o g/0000-0001-6499-0072 Email: apnicolas@unio i.es Jos´e Vi ´o ia Uni e si y o Coimb a Depa men o Ma hema ics La go D. Dinis 3000-143 Coimb a, Po ugal ORCID iD: h p://o cid.o g/0000-0003-3964-2425 Email: [email p o ec ed] SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C770