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
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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
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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
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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
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P. D. 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