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

DOI: 10.2478/auom-2023-0003 An. S¸t. Univ. Ovidius Constant¸a Vol. 31(1),2023,47–69 Skew-symmetric matrices related to the vector cross product in C7 P. D. Beites, A. P. Nicol´as, Jos´e Vit´oria Abstract Skew-symmetric matrices of order 7 defined through the 2-fold vector cross product in C7, and other related matrices, are presented. More concretely, matrix properties, namely invertibility, nullspace, powers and index, are studied. As a consequence, results on vector cross product equations, vector cross product differential equations and vector cross product difference equations in C7are established. 1 Introduction Assuming the usual definition, as explained by Elduque in the elementary account [13] on vector cross products and their connections with the exceptional basic classical simple Lie superalgebras, r-fold vector cross products exist only for d-dimensional vector spaces with: r= 1 and deven; r= 2 and d= 3 or 7; r= 3 and d= 8; and r=d−1 for an arbitrary d. The first proof of this classical result, and an extension of it, goes back to the work [7], where Brown and Gray presented an algebraic proof. An algebraic-topologic proof of the same result for real euclidean spaces was given by Eckmann, who in [12] assumed continuity – a weaker condition – instead of multilinearity. Based on the results in [12], a variation of the latter proof was given by Whitehead in [27]. In addition, in [16], citing the articles [12] and [27], Gray established Key Words: 2-fold vector cross product, Hermitian inner product, Skew-symmetric matrix, Generalized inverse, (Vector cross product, Vector cross product differential, Vector cross product difference) equation 2010 Mathematics Subject Classification: Primary 15A72; Secondary 15B57, 15A09. Received: 05.04.2022 Accepted: 15.07.2022 47 SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C748 results about vector cross products on manifolds. An elementary proof of the classical result, although only valid over a field of characteristic 0, was given by Rost in [24]. Later on, Meyberg simplified this proof in [23]. The mentioned classical result can be seen as a consequence of another classical result on the classification of Hurwitz algebras (that is, unital composition algebras, [2], [15], [18]). The real and complex cases are due to Hurwitz, who presented the classification in [19]. Jacobson established the classification, in [21], over a field Fof characteristic different from 2. More concretely, the generalized Hurwitz Theorem asserts that, over F, if Ais a finite dimensional composition algebra with identity, then its dimension is equal to 1, 2, 4 or 8. Furthermore, as Jacobson was interested in the study of the automorphisms of Hurwitz algebras, he proved that Ais isomorphic either to the base field, a separable quadratic extension of the base field (a quadratic commutative and associative separable algebra), a generalized quaternion algebra (a fourdimensional algebra that is associative but not commutative) or a generalized octonion algebra (also called Cayley algebra: an eight-dimensional algebra that is alternative but not associative), [21]. Throughout the years, the interest in 2-fold vector cross products has remained alive. In [20], Ikramov studies the complex vector cross product in C3. Costa, Facas Vicente, Beites, Martins, Serˆodio and Tadeu, in [10], use the vector cross product in R7to study the orthogonal projection of a point onto a line. In [9], Catarino and Vit´oria express the distance between two skew lines in R7in terms of the double vector cross product. In [3], vector cross product differential and difference equations are studied by Beites, Nicol´as, Saraiva and Vit´oria. A generalization of the standard definition of 2-fold vector cross product is proposed in [22] by Lewintan. In [5], Beites, Nicol´as and Vit´oria pursue an arithmetic for closed balls in Rnwhich includes operations involving the 2-fold vector cross product. Using this product in R3, Beites and Catarino establish Gelin-Ces`aro’s identity for Leonardo quaternions in [1]. Ferreira, Kaygorodov and Kudaybergenov describe derivations of complex Filippov algebras whose realizations generalize the 3-dimensional 2-fold vector cross product, [14]. The structure of the present work, divided into three main sections, is as follows. In section 2, where some background is presented, known definitions, results and notations related to the 2-fold vector cross product, to the 7-dimensional complex vector space C7, to generalized inverses and to differential and difference equations are recalled. In section 3, properties of matrices related to the 2-fold vector cross product in C7, namely on invertibility, nullspace, powers and index, are established. Partially following the ideas of Agudo for R3in [11], where he uses the term “vector division”, vector cross product equations in C7are considered in section 4. Moreover, in C7, SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C749 vector cross product differential equations and vector cross product difference equations are studied. Several results presented in the works [3] – of Beites, Nicol´as, Saraiva and Vit´oria –, [4] – due to Beites, Nicol´as and Vit´oria –, [17] – whose authors are Gross, Trenkler and Troschke –, [25] – of Trenkler –, and [26] – by Trenkler and Trenkler – are extended. 2 Preliminaries Let Vbe a d-dimensional vector space over a field Fof characteristic different from 2, endowed with a nondegenerate symmetric bilinear form (·,·). A bilinear map ×:V2→Vis a 2-fold vector cross product in Vif, for any u, v ∈V: 1. (u×v, u) = (u×v, v) = 0, 2. (u×v, u ×v) =  (u, u) (u, v) (v, u) (v, v) . Recall that 1. implies the skew-symmetry of the trilinear map (· × ·,·), and so the anticommutativity of ×, [13]. In the present article, the 2-fold vector cross product in the 7-dimensional complex vector space C7, denoted by ×, is considered. Equip the 7-dimensional complex vector space C7with the standard Hermitian inner product h·,·i :C7×C7→Cdefined by hx, yi= 7 X t=1 xtyt, for all x=x1. . . x7T,y=y1. . . y7T∈C7. It satisfies, respectively, linearity in the first coordinate, Hermitian (or conjugate) symmetry and positive definiteness: hαx +βy, zi=αhx, zi+βhy, zi,(1) hx, yi=hy, xi,(2) hx, xi ≥ 0 and hx, xi= 0 ⇔x= 0.(3) Recall that (2) implies that hx, xi ∈ R. Recall also that (1) and (2) imply conjugate linearity in the second coordinate, that is, hx, αy +βzi=αhx, yi+βhx, zi.(4) SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C750 When considering the 2-fold vector cross product in the 7-dimensional complex vector space C7, observe that the nondegenerate symmetric bilinear form (·,·) referred in the first definition is defined by (x, y) = hx, yi, for all x=x1. . . x7T,y=y1. . . y7T∈C7. Throughout the work, Cm×ndenotes the set of all m×ncomplex matrices. When n= 1, Cm×1is identified with Cm. When m=n= 1, C1×1is identified with C. Let B∈Cm×n. A matrix B(1) ∈Cn×mis a generalized inverse of Bif BB(1)B=B. See [6] for more details on generalized inverses, also known as (1)-inverses or g-inverses, where the subsequent result appears. Theorem 1 ([6]).Let B∈Cm×n,b∈Cm. Then, the equation Bx =bis consistent if and only if, for some B(1),BB(1)b=b. Let A∈Cn×n. The index Ind(A) of Ais the smallest l∈N0such that R(Al) = R(Al+1) or, equivalently, N(Al) = N(Al+1), where Rand Nstand for the column space (or range) and the nullspace, [8]. Alternatively, but equivalently, it can also be defined as the smallest l∈N0such that Cn=R(Al)⊕N(Al). Let Ind(A) = l. The Drazin inverse of Ais the unique matrix AD∈Cn×n which satisfies AAD=ADA, ADAAD=AD, Al+1AD=Al. When Ind(A)∈ {0,1},ADis sometimes called the group-inverse of Aand the last equality assumes the form AADA=A. There are several methods for computing AD, as described in [8] and references therein, some of which require all eigenvalues to be well determined. Let A, B ∈Cn×nand t0∈R. Let f=f(t) be a Cn-valued function of the real variable t. Throughout the work, x=x(t) stands for an unknown Cnvalued function of the real variable tand ˙x=dx dt denotes the corresponding derivative vector of x. A vector x0∈Cnis a consistent initial vector for the differential equation A˙x+Bx =f(5) if the initial value problem A˙x+Bx =f, x(t0) = x0,(6) possesses at least one solution. In this case, x(t0) = x0is said to be a consistent initial condition. Further, (5) is called tractable if (6) has a unique solution for each consistent initial vector x0, [8]. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C751 Theorem 2. [8] Let A, B ∈Cn×n. The homogeneous differential equation A˙x+Bx = 0 is tractable if and only if (λA +B)−1exists for some λ∈C. Let A, B ∈Cn×n. Let f(k)=f(k)(t)∈Cnbe the k-th term of a sequence of vectors, k= 0,1,2, .... Throughout the work, x(k)=x(k)(t)∈Cnstands for the k-th term of an unknown sequence of vectors, k= 0,1,2, . . . We assume that x(0) =x0is given. A vector x0∈Cnis a consistent initial vector for the difference equation Ax(k+1) =Bx(k)+f(k)(7) if the initial value problem Ax(k+1) =Bx(k)+f(k), k = 1,2, . . . , x(0) =x0,(8) has a solution for x(k). In this case, x(0) =x0is said to be a consistent initial condition. Furthermore, (7) is called tractable if (8) has a unique solution for each consistent initial vector x0, [8]. Theorem 3. [8] Let A, B ∈Cn×n. The homogeneous difference equation Ax(k+1) =Bx(k)is tractable if and only if (λA +B)−1exists for some λ∈C. 3 Properties Let a=a1a2a3a4a5a6a7T∈C7. Consider the linear mapping a×:C7→C7 x7→ a×(x) = a×x. For each a∈C7, there exists a unique matrix Sa∈C7×7such that a×x=Sax, (9) where 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 the following result, some properties related to the matrices defined in (9)-(10) are established. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C752 Proposition 4. Let a, b, c ∈C7. Let α, β ∈C. Then: 1. Sαa+βbc=αSac+βSbc; 2. Sa=Sa; 3. Sa=−ST a; 4. S∗ a=−Sa, where ·∗stands for the conjugate transpose of a matrix; 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 singular; 10. S2 a=aaT− ha, aiI7; 11. S3 a=−ha, aiSa; 12. the eigenvalues of Saare 0,p|ha, ai|eiθ 2and p|ha, ai|ei(θ 2+π), with θan argument of −ha, ai; 13. the nullspace of Sa, where a6= 0, is N(Sa) = {αa :α∈C}. Proof. Properties 1. and 5. are direct consequences of, respectively, the bilinearity and the anticommutativity of ×in C7. From (10) it is straightforward to prove 2. and 3. Concerning 4., invoking 2. and 3. leads to S∗ a= (Sa)T= (Sa)T=−Sa. Taking b=ain 5. leads to 6. By property 5., Saa+Saa= 0 which, by 2., is equivalent to Saa+Saa= 0⇔Saa+Saa= 0. The last equality means that each entry of Saais either zero or a purely imaginary complex number. Concretely, from (10), Saais the matrix 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)           , from where 7. follows. Applying 2. allows to arrive at 8. since Sab=Sab. As far as 9., on the one hand, if a= 0 then Sa= 0, a singular matrix. On the other hand, if a6= 0 then, from 6., Saa= 0. If Sawere invertible then a= 0, a contradiction. As S2 a= [sij]7×7with sij =       − 7 X t=1, t6=i a2 tif i=j aiajif i6=j , aaT= [dij]7×7with dij =a2 iif i=j aiajif i6=j, and ha, ai= 7 X t=1 a2 t, then 10. follows. Taking into account 10., S3 a=aaTSa− ha, aiSa. By 3. and 6., aaTSa= a(ST aa)T=−a(Saa)T= 0. Hence, 11. follows. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C754 Regarding 12., the characteristic equation of Sais det(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), with θan argument of −ha, ai. Let a∈C7\{0}. The inclusion ⊇in 13. follows from property 6. since, for all γ∈C,Sa(γa) = γSaa= 0. By the proof of 12., the eigenvalue 0 has algebraic multiplicity 1. As 0 6=a∈N(Sa), the geometric multiplicity of 0 is 1. Hence, dim N(Sa) = dim {αa :α∈C}= 1, and 13. is obtained. The subsequent results concern powers and traces of the matrices defined in (9)-(10). Lemma 5. Let a∈C7such that ha, ai 6= 0. For m∈N, S2m a= (−1)m+1ha, aim−1aaT+ (−1)mha, aimI7(11) and S2m+1 a= (−1)mha, aimSa.(12) Proof. The proof goes by induction on m. For (11), by 10. in Proposition 4, the base case holds. Also from 10. in Proposition 4 and the induction hypothesis, we have 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 the induction step holds too. For (12), by 11. in Proposition 4, it is straighforward to see that the base case holds. As for the induction step, by 10. in Proposition 4 and the induction hypothesis, we obtain S2m+3 a=S2m+1 aS2 a = (−1)mha, aimSa(aaT− ha, aiI7) = (−1)mha, aim(Saa)aT+ (−1)m+1ha, aim+1Sa. From here, taking into account 6. in Proposition 4, the second part of the result follows. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C755 Theorem 6. Let a∈C7such that ha, ai 6= 0. For m∈N,tr(S2m+1 a) = 0 and tr(S2m a) = 6(−1)mha, aim.(13) Proof. From (12) in Lemma 5, it is clear that tr(S2m+1 a)=(−1)mha, aimtr(Sa) = 0. From (11) in Lemma 5, taking into account aaTwritten for the proof of 10. in Proposition 4, tr(S2m a)=(−1)m+1ha, aim−1tr(aaT)+(−1)mha, aimtr(I7) =−(−1)mha, aim+ 7(−1)mha, aim, and the expression for the trace of S2m ain (13) is obtained. The following results are devoted to generalized inverses, invertibility and inverses of matrices related to the matrices defined in (9)-(10). Theorem 7. Let a∈C7such that ha, ai 6= 0. A generalized inverse of Sais S(1) a=−ha, ai−1Sa.(14) Proof. With ha, ai 6= 0, 11. in Proposition 4 leads to (14) since Sa−ha, ai−1SaSa=−ha, ai−1S3 a=Sa. Proposition 8. Let a, b ∈C7and γ∈C. The matrix γSa+Sbis singular. Proof. As Saand Sbare skew-symmetric matrices, then, for any γ∈C,γSa+ Sbis also skew-symmetric of odd order. Hence, det(γSa+Sb) = 0. Lemma 9. Let a∈C7and α∈C. The matrix Sa+αI7is non-singular if and only if α6= 0 and αis not a square root of −ha, ai. Proof. A straightforward calculation of det(Sa+αI7) leads to α(α2+ha, ai)3. In the stated conditions, det(Sa+αI7) = 0 if and only if α= 0 or α2= −ha, ai. Theorem 10. Let a∈C7. Let α∈C\{0}such that αis not a square root of −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 independent of the used λ. Hence, in what follows, we drop the subscripts λ and take λ= 0. From Theorem 12, Ind( ˆ Sa) = 1. Invoking [8, Theorem 9.2.3, p. 175], we obtain the necessary and sufficient condition x0∈R(ˆ Sa) = R(ˆ SD aˆ Sa) for a vector x0∈C7to be a consistent initial vector for (27). Since ˆ SD aˆ Sa=ˆ Saˆ SD a, we get (28). As ˆ Sa=B−1Sa, then, by (15) in Theorem 10, we obtain (29). Assume now that x0∈C7is a consistent initial vector for (27). As ˆ B=I7, once again from [8, Theorem 9.2.3], the unique solution of the homogeneous initial value problem Sa˙x+Bx = 0, x(t0) = x0, is given by (30). Theorem 27. Let a∈C7with ha, ai 6= 0,b∈C7\{0}and α∈C\{0}such that αis not a square root of −hb, bi. Let f=f(t)be a C7-valued function of the real variable t, continuously differentiable around t0, and let x=x(t) an unknown C7-valued function of the real variable t. A vector x0∈C7is a consistent initial vector for the vector cross product differential equation a×˙x+b×x+αx =f(31) if and only if x0is of the form x0= (I−ˆ Saˆ SD a)ˆ f(t0) + ˆ Saˆ SD aq, (32) for some vector q∈C7, where ˆ Sa=−(α2+hb, bi)−1Sb−αI7−α−1bbTSa(33) and ˆ f=−(α2+hb, bi)−1Sb−αI7−α−1bbTf. (34) Moreover, if x0∈C7is a consistent initial vector for (31), then the unique solution of (31), with initial condition x(t0) = x0, is x(t) = e−ˆ SD a(t−t0)ˆ Saˆ SD ax0+e−ˆ SD atZt t0 eˆ SD asˆ SD aˆ f(s)ds+(I7−ˆ Saˆ SD a)ˆ f(t).(35) Proof. By (9), we can rewrite equation (31) as Sa˙x+ (Sb+αI7)x=f, where α∈C\{0}is such that α26=−hb, bi. As in the proof of Theorem 26, let B=Sb+αI7,ˆ Sa=B−1Sa,ˆ B=I7,ˆ f=B−1f. Taking into account Theorem 12, Ind( ˆ Sa) = 1. The necessary and sufficient condition x0∈ {(I7−ˆ Saˆ SD a)ˆ f(t0) + R(ˆ SD aˆ Sa)}for a vector x0∈C7to be a consistent initial vector for (31) comes from [8, Theorem 9.2.3, p. 175], which leads to (32). By (15) in Theorem 10, we obtain (33) and (34). Suppose now that x0∈C7is a consistent initial vector for (31). Once again from [8, Theorem 9.2.3], the unique solution of the inhomogeneous initial value problem Sa˙x+Bx =f, x(t0) = x0, is given by (35). SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C763 4.3 Vector Cross Product Difference Equations In the present section, some vector cross product difference equations in C7 are studied. Theorem 28. Let b∈C7such that hb, bi 6= 0 and let x(k)∈C7be the k-th term of an unknown sequence of vectors, k= 0,1,2, ... The unique solution of the vector cross product difference equation x(k+1) =b×x(k),(36) with initial condition 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, even (37) where β=|hb, bi|1/2eiθ 2, with θan argument of hb, bi. Proof. Due to (9), equation (36) assumes the form x(k+1) =Sbx(k), which is a tractable equation by Theorem 3. In fact, from Lemma 9, (λI7+Sb)−1exists for every λ∈C\{0}that is not a square root of −hb, bi. Taking into account the recurrence relation, the unique solution of the homogeneous initial value problem x(k+1) =Sbx(k),k= 0,1,2, . . . ,x(0) =x0, is given by x(k)=Sk bx0, k = 0,1,2, ... From Lemma 5, we arrive at (37). Theorem 29. Let b∈C7such that hb, bi 6= 0. Let f(k)∈C7be the k-th term of a sequence of vectors, k= 0,1,2, ..., and let x(k)∈C7be the k-th term of an unknown sequence of vectors, k= 0,1,2, .... The unique solution of the vector cross product difference equation x(k+1) =b×x(k)+f(k),(38) with initial condition x(0) =x0, is x(k)=                x0, k = 0 (−1)k−1 2βk−1Sbx0+ k−1 X i=0 Sk−1−i bf(i), k ∈N, odd (−1)k 2+1βk−2bbT+ (−1)k 2βkI7x0+ k−1 X i=0 Sk−1−i bf(i), k ∈N, even (39) where β=|hb, bi|1/2eiθ 2, with θan argument of hb, bi. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C764 Proof. Again by (9), equation (38) assumes the form x(k+1) =Sbx(k)+f(k). The recurrence relation allows to obtain the unique solution of the inhomogeneous initial value problem x(k+1) =Sbx(k)+f(k),k= 0,1,2, . . . ,x(0) =x0, given by x(k)=Sk bx0+ k−1 X i=0 Sk−1−i bf(i), k = 1,2, ... (40) From Lemma 5, we obtain (39). Corollary 30. Let b∈C7such that hb, bi 6= 0,c∈C7and let x(k)∈C7be the k-th term of an unknown sequence of vectors, k= 0,1,2, .... The unique solution of the vector cross product difference equation x(k+1) =b×x(k)+c, (41) with initial condition 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, even (42) where β=|hb, bi|1/2eiθ 2, with θan argument of hb, bi. Proof. A particular case of the previous result, putting cinstead of the sequence f(k)k∈N0. Theorem 31. Let a, b ∈C7\{0}and let x(k)∈C7be the k-th term of an unknown sequence of vectors, k= 0,1,2, ... The vector cross product difference equation a×x(k+1) =b×x(k)(43) is not tractable. Proof. From (9), the rewriting of equation (43) leads to Sax(k+1) =Sbx(k). From Proposition 8, for any λ∈C,λSa+Sbis a singular matrix and the result follows from Theorem 3. Similarly to subsection 4.2, due to the previous result, perturbed versions of the difference equation (43) are now studied. SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C765 Theorem 32. Let a∈C7with ha, ai 6= 0,b∈C7\{0}and α∈C\{0}such that αis not a square root of −hb, bi. Let x(k)∈C7be the k-th term of an unknown sequence of vectors, k= 0,1,2, .... A vector x0∈C7is a consistent initial vector for the vector cross product difference equation a×x(k+1) =b×x(k)+αx(k)(44) if and only if x0is of the form x0=ˆ Saˆ SD aq, (45) for some q∈C7, where ˆ Sa=−(α2+hb, bi)−1Sb−αI7−α−1bbTSa.(46) Moreover, if x0∈C7is a consistent initial vector for (44), then the unique solution of (44), with initial condition x(0) =x0, is x(k)=ˆ SD ak x0, k = 0,1,2, . . . (47) Proof. From (9), equation (44) assumes the form Sax(k+1) =Bx(k)where B=Sb+αI7with α∈C\{0}such that αis not a square root of −hb, bi. By Lemma 9, Bis non-singular. Owed to this fact, λSa+Bis also a non-singular matrix if λ= 0 and, by Theorem 3, (44) is a tractable equation. Following the notation in [8], let ˆ Sa,λ = (λSa+B)−1Saand ˆ Bλ= (λSa+B)−1B, where λ∈Cis such that λSa+Bis non-singular. By [8, Theorem 9.2.2, p. 174], the consistency of an initial vector for (44) and its general solution are independent of the used λ. Hence, in what follows, we drop the subscripts λ and take λ= 0. By Theorem 12, Ind( ˆ Sa) = 1. Invoking [8, Theorem 9.3.2, p. 182-183], we get the necessary and sufficient condition x0∈R(ˆ Sa) = R(ˆ SD aˆ Sa) for a vector x0∈C7to be a consistent initial vector for (44). As ˆ SD aˆ Sa=ˆ Saˆ SD a, we obtain (45). Since ˆ Sa=B−1Sa, then, by (15) of Theorem 10, we arrive at (46). Suppose now that x0∈C7is a consistent initial vector for (44). Since ˆ B= I7, once again from [8, Theorem 9.3.2], the unique solution of the homogeneous initial value problem Sax(k+1) =Bx(k),k= 0,1, . . . ,x(0) =x0, is given by (47). Theorem 33. Let a∈C7with ha, ai 6= 0,b∈C7\{0}and α∈C\{0}such that αis not a square root of −hb, bi. Let f(k)∈C7be the k-th term of SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C766 a sequence of vectors, k= 0,1,2, ..., and let x(k)∈C7the k-th term of an unknown sequence of vectors, k= 0,1,2, .... A vector x0∈C7is a consistent initial vector for the vector cross product difference equation a×x(k+1) =b×x(k)+αx(k)+f(k), k = 0,1,2,..., (48) if and only if x0is of the form x0=−I7−ˆ Saˆ SD aˆ f(0) +ˆ Saˆ SD aq, (49) for some q∈C7, where ˆ Sa=−(α2+hb, bi)−1Sb−αI7−α−1bbTSa(50) and ˆ f(k)=−(α2+hb, bi)−1Sb−αI7−α−1bbTf(k).(51) Moreover, if x0∈C7is a consistent initial vector for (48), then the unique solution of (48), with initial condition x(0) =x0, is x(k)given 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 f(i)−I7−ˆ Saˆ SD aˆ f(k), k = 1,2, . . . (52) Proof. By (9), the rewriting of equation (48) leads to Sax(k+1) =Bx(k)+f(k), where B=Sb+αI7with α∈C\{0}such that α26=−hb, bi. As in the proof of Theorem 32, let ˆ Sa=B−1Sa,ˆ B=I7,ˆ f(k)=B−1f(k). From Theorem 12, Ind( ˆ Sa) = 1. The necessary and sufficient condition x0∈ {−(I7−ˆ Saˆ SD a)ˆ f(0) +R(ˆ SD aˆ Sa)}for a vector x0∈C7to be a consistent initial vector for (48) comes from [8, Theorem 9.3.2, p. 182-183]. Thus, we obtain (49). By (15), we get (50) and (51). Assume now that x0∈C7is a consistent initial vector for (48). Once again from [8, Theorem 9.3.2], the unique solution of the inhomogeneous initial value problem Sax(k+1) =Bx(k)+f(k),k= 0,1,2, . . . ,x(0) =x0, is given by (52). Acknowledgment P. D. Beites was supported by FCT (Funda¸c˜ao para a Ciˆencia e a Tecnologia, Portugal), research project UIDB/00212/2020 of CMA-UBI (Centro de Matem´atica e Aplica¸c˜oes, Universidade da Beira Interior, Portugal), and by the research project MTM2017-83506-C2-2-P, Spain. The author A. P. 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Nicol´as Departamento de Matem´aticas Universidad de Oviedo Calle Federico Garc´ıa Lorca, 18 33007 Oviedo, Espa˜na ORCID iD: https://orcid.org/0000-0001-6499-0072 Email: [email protected] Jos´e Vit´oria University of Coimbra Department of Mathematics Largo D. Dinis 3000-143 Coimbra, Portugal ORCID iD: http://orcid.org/0000-0003-3964-2425 Email: [email protected] SKEW-SYMMETRIC MATRICES RELATED TO THE VECTOR CROSS PRODUCT IN C770