# On Generalized Rotation Matrices

Authors: Oskar Maria Baksalary; Götz Trenkler
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

A general class of matrices, covering, for instance, an important set of proper rotations, is considered. Several characteristics of the class are established, which deal with such notions and properties as determinant, eigenspaces, eigenvalues, idempotency, Moore-Penrose inverse, or orthogonality.


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## Body

## 1. Introduction and Basic Properties

Letℂm,n denote the set of m×n complex matrices. The symbols L′, L*, ℛ(L), and rk(L) will stand for the transpose, conjugate transpose, column space, and rank, respectively, of L∈ℂm,n. Further, L†∈ℂn,m will be the Moore-Penrose inverse of L∈ℂm,n, that is, the unique matrix satisfying the equations
(1.1)LL†L=L,L†LL†=L†,(LL†)*=LL†,(L†L)*=L†L,
and In will mean the identity matrix of order n. The Moore-Penrose inverse L† is useful in representing the orthogonal (in the sense of the standard inner product) projectors onto ℛ(L) and ℛ(L*), denoted by PL and PL*, as well as the orthogonal projectors onto the orthogonal complements of these subspaces, denoted by QL and QL*. To be precise, for L∈ℂm,n,
(1.2)PL=LL†,PL*=L†L,QL=Im-LL†,QL*=In-L†L.
With respect to a scalar, say α∈ℂ, the inverse α† is defined as: α†=0 when α=0 and α†=α-1 when α≠0.The considerations of the present paper concern3×3 matrices and 3×1 vectors having either complex or real entries; in the latter case their subsets will be denoted by ℝ3,3 and ℝ3,1, respectively. Customarily, the symbol ∥l∥ will stand for the Euclidean norm of l∈ℂ3,1, that is, ∥l∥=l*l. Let
(1.3)Ta=(0-a3a2a30-a1-a2a10)
be generated by a=(a1,a2,a3)′∈ℝ3,1. The matrix Ta can be used to define the vector cross product in ℝ3,1, with Tal=a×l for any l∈ℝ3,1; see [1]. Further properties of Ta are listed in the following lemma, whose proof is easy and thus omitted.Lemma 1.1.
Letλ,μ∈ℝ and  a,b,c,d∈ℝ3,1. Moreover, let  Tl for some  l∈ℝ3,1 be of the form  (1.3). Then,  (i)
Tλa+μb=λTa+μTb,(ii)
Tab=-Tba,(iii)
Ta=-Ta′,(iv)
Taa=0,(v)
TaTb=ba′-(a′b)I3,(vi)
(TaTb)′=TbTa,(vii)
TaTbTc=Tacb′-(b′c)Ta,(viii)
TaTbTa=-(a′b)Ta,(ix)
Ta3=-∥a∥2Ta,(x)
c′Tab=a′Tbc=b′Tca,(xi)
TaTbc=(a′c)b-(a′b)c=a×(b×c),(xii)
(Tab)′Tcd=(a′c)(b′d)-(a′d)(b′c)=(a×b)′(c×d),(xiii)
TTab=ba′-ab′=TaTb-TbTa.Relationships listed in Lemma1.1 are available in the literature; see Trenkler [2, 3], Groß et al. [4], Bernstein [5, Ch. 3], and G. Trenkler and D. Trenkler [6]. It is noteworthy that the three scalars involved in point (x) represent the scalar triple products, for example, c′Tab=(a×b)′c. Moreover, the right-hand sides equalities in points (xi) and (xii) are known as the Grassmann and Lagrange identities, respectively. From the point of view of the present paper, conditions (iii)–(v) of the lemma are of particular importance and will be extensively utilized in the subsequent derivations.It is known that ifa∈ℝ3,1 generating Ta given in (1.3) is such that ∥a∥=1, then the matrices of the form
(1.4)R=sinθTa+I3+(1-cos⁡θ)Ta2
describe proper rotations in ℝ3,1 with θ being the angle of rotation about the axis given by a; see Noble [7, Ch. 12] or Murray et al. [8, Ch. 2]. In what follows we consider a more general class of matrices than the one spanned by matrices of the form (1.4), namely,
(1.5)Υ={T∈C3,3:T=αTa+βI3+γaa′,α,β,γ∈C,a∈R3,1,a≠0}.
It can be verified that Υ covers all proper rotations of the form (1.4). Moreover, it comprises also improper rotations, that is, orthogonal matrices with determinant equal to −1 [9, Ch. VIII], symmetric elementary matrices [10, Sec.  1], and all matrices commuting with Ta [11].The purpose of the present paper is to identify various properties of the class of matrices specified in (1.5). As a result, several characteristics of the class Υ are established, dealing with such notions and properties as idempotency, determinant, eigenvalues, Moore-Penrose inverse, orthogonality, or eigenspaces.


## 2. Results

Subsequently, the symboli is interpreted as i=-1. The theorem below states that Υ is closed against multiplication.Theorem 2.1.
LetT1,T2∈Υ with  Tk=αkTa+βkI3+γkaa′,  k=1,2. Then  T1T2∈Υ.Proof.
Direct calculations show that(2.1)T1T2=(α1β2+β1α2)Ta+(β1β2-α1α2‖a‖2)I3+(α1α2+β1γ2+β2γ1+γ1γ2‖a‖2)aa′,
establishing the assertion.It is clear that, besides ensuring the closure property, multiplication inΥ is also associative. Furthermore, for each T∈Υ there exists the identity element, namely, I3. On the other hand, since Υ includes also singular matrices, not every T∈Υ has an inverse element in Υ. Thus, the set Υ is a semigroup under (matrix) multiplication. (As will be seen subsequently, the Moore-Penrose inverse of every T∈Υ belongs to Υ, in particular, T∈Υ⇔T-1∈Υ for each nonsingular T∈Υ.)The next theorem provides necessary and sufficient conditions forT2=T.Theorem 2.2.
LetT∈Υ. Then  T is idempotent if and only if
(2.2)2αβ=α,β2-α2‖a‖2=β,α2+2βγ+γ2‖a‖2=γ.Proof.
The equivalence established in the theorem follows straightforwardly from Theorem2.1.In view of Theorem2.2, we can distinguish two subsets of idempotent matrices belonging to Υ corresponding to α≠0 and α=0. In the former of them, α=±(i/2∥a∥), β=1/2, and γ=±(1/2∥a∥2). Hence, since a†=(1/∥a∥2)a′, it follows that T∈{P1,P2,P3,P4}, with
(2.3)P1=12(i‖a‖Ta+I3+Pa),P2=12(i‖a‖Ta+Qa),P3=12(-i‖a‖Ta+I3+Pa),P4=12(-i‖a‖Ta+Qa).
(Note that matrices Pk, k=1,…,4, were considered by Trenkler [3] to characterize certain eigenspaces.) On the other hand, if α=0, then Theorem 2.2 entails four cases, namely, β=0, γ=0; β=0, γ=1/∥a∥2; β=1, γ=0; and β=1, γ=-1/∥a∥2, leading to P5=0, P6=Pa, P7=I3, and P8=Qa, respectively.The next task is to characterize the eigenvalues of matrices belonging toΥ. The subsequent theorem expresses the determinant of T∈Υ in terms of the scalars α, β, and γ. Its proof is based on the so-called Leverrier-Sourian-Frame algorithm, which provides a useful tool to calculate the coefficients in a characteristic polynomial. Since the algorithm is not widely known, it is restated in the following lemma; see for example, Meyer [12, page 504]. Customarily, tr(·) denotes the trace of a matrix argument.Lemma 2.3.
LetL∈ℂn,n, and let  μn+c1μn-1+c2μn-2+⋯+cn=0 be the characteristic equation for  L. Then
(2.4)c1=-tr⁡(L),ck=-1ktr⁡(LBk-1),k=2,3,…,n,
where  B1=c1In+L and  Bk=ckIn+LBk-1,  k=2,3,…,n-1.Theorem 2.4.
LetT∈Υ, and let  τ=α2∥a∥2+β2. Then  det(T)=τ(β+γ∥a∥2).Proof.
From Lemma2.3 it follows that c1=-tr(T) is given by c1=-(3β+γ∥a∥2), whence it is seen that B1=c1I3+T takes the form
(2.5)B1=αTa-(2β+γ‖a‖2)I3+γaa′.
Further, if k=2, then c2=-(1/2)tr(TB1), where
(2.6)TB1=-(αβ+αγ‖a‖2)Ta-(α2‖a‖2+2β2+βγ‖a‖2)I3+(α2-βγ)aa′.
Since tr(Ta)=0, in consequence we get c2=α2∥a∥2+3β2+2βγ∥a∥2, leading to
(2.7)B2=c2I3+TB1=-(αβ+αγ‖a‖2)Ta+(β2+βγ‖a‖2)I3+(α2-βγ)aa′.
Finally, if k=3, then c3=-(1/3)tr(TB2), where
(2.8)TB2=(α2β‖a‖2+α2γ‖a‖4+β3+β2γ‖a‖2)I3.
Hence, straightforward calculations yield c3=-τ(β+γ∥a∥2). Combining this result with the property det(T)=(-1)3c3 completes the proof.By virtue of Theorem2.4, it is easy to determine the spectrum of matrices belonging to Υ.Theorem 2.5.
LetT∈Υ. Then the eigenvalues of  T are solutions  μ to the equation  [α2∥a∥2+(β-μ)2][β-μ+γ∥a∥2]=0.Proof.
It is clear that(2.9)det⁡(T-μI3)=det⁡[αTa+(β-μ)I3+γaa′].
Hence, on account of Theorem 2.4, we get
(2.10)det⁡(T-μI3)=det⁡{[α2‖a‖2+(β-μ)2][β-μ+γ‖a‖2]},
establishing the assertion.Theorem2.5 leads to what follows.Corollary 2.6.
LetT∈Υ. Then the eigenvalues of  T are
(2.11)μ1=β+γ‖a‖2,μ2=β+iα‖a‖2,μ3=β-iα‖a‖.An expected result originating from Corollary2.6 is that μ1μ2μ3=det(T), with det(T) given in Theorem 2.4. Furthermore, when α=1, β=0, and γ=0, then the eigenvalues given in (2.11) reduce to μ1=0, μ2=i∥a∥, and μ3=-i∥a∥, that is, to the eigenvalues of Ta; see [3, Theorem  2].The following theorem will be useful in the subsequent calculations of the Moore-Penrose inverses ofT∈Υ.Theorem 2.7.
LetA∈ℂ3,3 be such that  A=αTa+βI3, with  α,β∈ℂ and  Ta of form  (1.3) generated by nonzero  a∈ℝ3,1. Moreover, let  λ=1+γa′A†a, with  γ∈ℂ, and  τ=α2∥a∥2+β2. Then, (i)
det(A)=βτ,(ii)
the eigenvalues ofA are  σ1=β,  σ2=β+iα∥a∥,  σ3=β-iα∥a∥,(iii)
ifβ=0, then  A†=-(α†/∥a∥2)Ta,  λ=1,(iv)
ifβ≠0,  τ=0, then  A†=(1/4β)((α/β)Ta+I3+3Pa),  λ=1+(γ/β)∥a∥2,(v)
ifβ≠0,  τ≠0, then  A-1=(1/τ)(-αTa+βI3+(α2/β)aa′),  λ=1+(γ/β)∥a∥2.Proof.
Assertion (i) follows from Theorem2.4 by setting γ=0. Statement (ii) is a consequence of (i), whereas the validity of points (iii) and (v) can be confirmed by straightforward calculations; see [3, Theorem  1]. For the proof of statement (iv) note that β≠0, τ=0 imply α/β=±(i/∥a∥), that is, α/β is purely imaginary. Taking this fact into account, in view of a†=(1/∥a∥2)a′, the validity of the formula for A† given in point (iv) is seen by direct verification of conditions (1.1). Similarly, the formula for A-1 provided in point (v) can be confirmed by examining the condition AA-1=I3. The proof is thus complete, for the expressions for λ given in points (iv) and (v) are easily obtainable.Note that regardless whetherβ and/or τ in Theorem 2.7 are zero or not, the matrix A is such that PA=PA*, or, in other words, ℛ(A)=ℛ(A*), that is, A is an EP matrix. Another observation is that by setting α=1, β=1 in Theorem 2.7 leads to the relationship
(2.12)(Ta+I3)-1=1‖a‖2+1(-Ta+I3+aa′).
Hence, the so called Cayley transform of Ta (see [13, p. 219]), being of the form S=(-Ta+I3)(Ta+I3)-1, of which it is known that is orthogonal (see [9, Theorem  8.1.10]), takes the form
(2.13)S=1‖a‖2+1[-2Ta+(1-‖a‖2)I3+2aa′].
Since det(S)=1, the matrix S represents in fact a proper rotation.We now have the tools necessary to establish formulae for the Moore-Penrose inverses ofT∈Υ.Theorem 2.8.
LetT∈Υ be decomposed as  T=A+γaa′, where  A=αTa+βI3. Moreover, let  λ=1+γa'A†a and  τ=α2∥a∥2+β2. Then, (i)
ifβ=0, then  T†=(1/∥a∥2)(-α†Ta+γ†Pa),(ii)
ifβ≠0,  τ=0,  λ=0, then  T†=(1/4β)((α/β)Ta+Qa),(iii)
ifβ≠0,  τ=0,  λ≠0, then  T†=(1/4β)((α/β)Ta+I3+((3β-γ∥a∥2)/(β+γ∥a∥2))Pa),(iv)
ifβ≠0,  τ≠0,  λ=0, then  T†=(1/τ)(-αTa+βQa),(v)
ifβ≠0,  τ≠0,  λ≠0, then  T-1=(1/τ)(-αTa+βI3+((α2-βγ)/(β+γ∥a∥2))aa′).Proof.
The first observation is that ifβ=0, then on account of Theorem  3.1.1 in [14], we have T†=(αTa)†+(γaa′)†. Hence, by utilizing Ta†=-(1/∥a∥2)Ta and (aa′)†=(1/∥a∥2)Pa, the formula for T† given in point (i) follows.
Assume now thatβ≠0, in which case we can still have τ=0 or τ≠0. In the former of these situations, point (iv) of Theorem 2.7 implies PA=(1/2)((α/β)Ta+I3+Pa), whence PAa=a, or, in other words, a∈ℛ(A). This inclusion is clearly satisfied also when τ≠0, for then det(A)≠0.
In order to apply the results of Baksalary et al. in [15], we introduce b=γa, c=a. Then, T=A+bc′=A+bc*, that is, T is a rank-one modification of A. As in [15], we define also the vectors d,e,f,g∈ℂ3,1 according to
(2.14)d=A†b,e=(A†)*c,f=QAb,g=QA*c,
and denote the squares of the norms of the first two of them by δ and η, that is, δ=∥d∥2, η=∥e∥2. As is seen from Theorem 2.7, the scalar λ∈ℂ specified in [15] by λ=1+c*A†b now takes the form λ=1+γa′A†a=1+(γ/β)∥a∥2.
Let us first consider case (ii) of the theorem, characterized, in addition toβ≠0, by τ=0, λ=0. On account of Theorem  1.1 in [15] this case corresponds to rk(T)=rk(A)-1. As can be directly verified with the use of A† given in point (iv) of Theorem 2.7, d=(γ/β)a and e=(1/β¯)a, from where δ=|γ/β|2∥a∥2 and η=(1/|β|2)∥a∥2. Moreover, we get A†e=(1/|β|2)a, implying d*A†e=(γ¯/|β|2β¯)∥a∥2 and A†ee*=(1/|β|2β)aa′. Furthermore, dd*=(|γ/β|2)aa′ and de*=(γ/β2)aa′. Thus,
(2.15)δ-1dd*=Pa,η-1A†ee*=1βPa,δ-1η-1(d*A†e)de*=1βPa.
In consequence, from formula (2.1) in [15] we get T†=QaA†, whence the expression for T† claimed in point (ii) of the theorem follows.
According to Theorem  1.1 in [15], another case which corresponds to rk(T)=rk(A)-1 is given in point (iv) of the theorem, where τ≠0, λ=0. Direct calculations with the use of A-1 given in point (v) of Theorem 2.7 show that formulae (2.15) remain valid also in this case. Hence, from relationship (2.1) in [15] we obtain T†=QaA-1, leading to the expression claimed in point (iv) of the theorem.
Another conclusion originating from Theorem  1.1 in [15] is that case (iii) of the theorem, in which τ=0, λ≠0, corresponds to rk(T)=rk(A). Direct calculations with the use of A† given in point (iv) of Theorem 2.7 show that
(2.16)λ-1de*=γβ(β+γ‖a‖2)aa′,
and substituting this relationship into formula (2.2) in [15] leads to the expression for T† given in the theorem.
Case (v), in whichτ≠0, λ≠0, is left to be considered. According to the remark on p. 210 in [15], in such a situation T is nonsingular. Hence, T†=T-1, and the validity of the formula given in the theorem can be confirmed by direct verifications of the condition TT-1=I3.A conclusion originating from Theorem2.8 is that every T∈Υ satisfies T∈Υ⇔T†∈Υ, the property which was already mentioned in the remark following Theorem 2.1. Further consequences of Theorem 2.8 are in what follows.Corollary 2.9.
LetT∈Υ, and let  λ=1+γa′A†a,  τ=α2∥a∥2+β2. Then  PT=PT*, where (i)
ifβ=0, then  PT=αα†Qa+γγ†Pa,(ii)
ifβ≠0,  τ=0,  λ=0, then  PT=P2 provided that  α/β=i/∥a∥ and  PT=P4 provided that  α/β=-i/∥a∥,(iii)
ifβ≠0,  τ=0,  λ≠0, then  PT=P1 provided that  α/β=i/∥a∥ and  PT=P3 provided that  α/β=-i/∥a∥,(iv)
ifβ≠0,  τ≠0,  λ=0, then  PT=Qa,(v)
ifβ≠0,  τ≠0,  λ≠0, then  PT=I3,with  Pk,  k=1,…,4, as specified in (2.3).Proof.
The corollary is established by direct calculations.Point (v) of Theorem2.8 enables to formulate necessary and sufficient conditions for T∈Υ to be orthogonal.Theorem 2.10.
LetT∈Υ with nonzero  α,β,γ∈ℝ. Moreover, let  τ=α2∥a∥2+β2. Then  T∈Υ is orthogonal if and only if
(2.17)τ=1,(β+γ‖a‖2)2=1.Proof.
The matrixT is orthogonal if and only if it is nonsingular and T-1=T′. On account of point (v) of Theorem 2.8, it is seen that T-1=T′ is equivalent to
(2.18)α=ατ,β=βτ,γ=α2-βγβ+γ‖a‖2,
or, in other words, τ=1 and γ(β+γ∥a∥2)=α2-βγ. Taking into account that τ=1 implies α2=(1/∥a∥2)(1-β2), the assertion follows.Observe that the right-hand side condition in (2.17) admits two possibilities, namely, either β+γ∥a∥2=1 or β+γ∥a∥2=-1. Since τ=1, Theorem 2.4 ensures that in the former situation T is a proper rotation, whereas in the latter one T is an improper rotation.The next theorem concerns eigenspaces attributed toT∈Υ.Theorem 2.11.
LetT∈Υ, and let  ℰ(μj),  j=1,2,3, be the eigenspaces of  T associated with its eigenvalues  μj given in  (2.11). Then, (i)
ifα=0, then  ℰ(μ2)=ℛ(I3-γγ†Pa),  ℰ(μ3)=ℛ(I3-γγ†Pa),(ii)
ifα≠0, then  ℰ(μ2)=ℛ(Q2) provided that  γ/α=i/∥a∥ and  ℰ(μ2)=ℛ(Q1) otherwise and, simultaneously,  ℰ(μ3)=ℛ(Q4) provided that  γ/α=-(i/∥a∥) and  ℰ(μ3)=ℛ(Q3) otherwise(iii)
ifγ=0, then  ℰ(μ1)=ℛ(I3-αα†Qa),(iv)
ifγ≠0, then  ℰ(μ1)=ℛ(Q2) provided that  α/γ=-i∥a∥,  ℰ(μ1)=ℛ(Q4) provided that  α/γ=i∥a∥, and  ℰ(μ1)=ℛ(a) otherwise,where  Qk=I3-Pk,  k=1,…,4, with  Pk as specified in  (2.3).Proof.
It is known thatℰ(μj)=ℛ[I3-T(μj)T(μj)†], where T(μj)=T-μjI3, j=1,2,3; see, for example, [3]. Clearly, for each μj, matrix T(μj) can be written as
(2.19)T(μj)=αTa+β̃I3+γaa′,
where β̃=β-μj. For μ1=β+γ∥a∥2, we have β̃=-γ∥a∥2. By virtue of the equivalence β̃=0⇔γ=0, statement (i) of Corollary 2.9 leads to point (iii) of the theorem. If, however, β̃≠0, that is, γ≠0, then λ=1+(γ/β̃)∥a∥2=0, which means that cases (iii) and (v) of Corollary 2.9, characterized by λ≠0, are not attainable in the present situation. Further observations are that τ=α2∥a∥2+β̃2=(α2+γ2∥a∥2)∥a∥2 and α/β̃=-(α/γ)(1/∥a∥2). In view of these facts, it is seen that statements (ii) and (iv) of Corollary 2.9 lead to characterizations of ℰ(μ1) given in point (iv) of the theorem.
Next we consider eigenvalueμ2=β+iα∥a∥, which ensures that β̃ occurring in (2.19) is given by β̃=-iα∥a∥. Since β̃=0 is equivalent to α=0, on account of statement (i) of Corollary 2.9 we arrive at characterization of ℰ(μ2) given in point (i) of the theorem. On the other hand, if β̃≠0, that is, α≠0, then τ is necessarily equal to zero, which means that cases (iv) and (v) of Corollary 2.9 are to be excluded from the present considerations. Furthermore, it is seen that λ=1+i(γ/α)∥a∥ and α/β̃=i/∥a∥. With these facts taken into account, we conclude that statements (ii) and (iii) of Corollary 2.9 lead to characterizations of ℰ(μ2) provided in point (ii) of the theorem.
The last eigenvalue to be considered isμ3=β-iα∥a∥, for which β̃=iα∥a∥. In this case, analogous arguments to the ones used with respect to μ2 lead to the eigenspaces ℰ(μ3) in points (i) and (ii) of the theorem. The proof is complete.Observe that ifα=1, β=0, and γ=0, then from Theorem 2.11 we get ℰ(μ1)=ℛ(a), ℰ(μ2)=ℛ(Q1), and ℰ(μ3)=ℛ(Q3), that is, the eigenspaces of Ta identified by Trenkler [3, Sec. 3].Theorem2.11 is supplemented with examples demonstrating its applicability. Let a=(1,0,-1)′. Then, (2.20)Ta=(010-10-1010),aa′=(10-1000-101),
leading to
(2.21)T=(β+γα-γ-αβ-α-γαβ+γ)
of eigenvalues μ1=β+2γ, μ2=β+i2α, μ3=β-i2α. From the right-hand side formula in (2.20) we get
(2.22)Pa=(120-12000-12012),
and the projectors Qk, k=1,…,4, involved in Theorem 2.11 are of the forms
(2.23)Q1=(14-i2414i2412i2414-i2414),Q2=(34-i24-14i2412i24-14-i2434),Q3=(14i2414-i2412-i2414i2414),Q4=(34i24-14-i2412-i24-14i2434).
Hence, from Theorem 2.11 we obtain what follows.Ifα=0, then ℰ(μ2)=ℰ(μ3)=ℂ3,1 provided that γ=0 and
(2.24)E(μ2)=E(μ3)=span⁡{(101),(010)}
otherwise.Next, ifα≠0, then
(2.25)E(μ2)=span⁡{(10-1),(01-i2)}
provided that γ/α=i(2/2) and
(2.26)E(μ2)=span⁡{(1i21)}
otherwise and, simultaneously,
(2.27)E(μ3)=span⁡{(10-1),(01i2)}
provided that γ/α=-i(2/2) and
(2.28)E(μ3)=span⁡{(1-i21)}
otherwise.Further, ifγ=0, then ℰ(μ1)=ℂ3,1 provided that α=0 and ℰ(μ1)=ℛ(a) otherwise.Finally, ifγ≠0, then
(2.29)E(μ1)=span⁡{(10-1),(01-i2)}
provided that α/γ=-i2,
(2.30)E(μ1)=span⁡{(10-1),(01i2)}
provided that α/γ=i2, and ℰ(μ1)=ℛ(a) otherwise.Further consequences of Theorem2.11 deal with the proper rotation matrices; for more detailed discussion see [16]. G. Trenkler and D. Trenkler [6, Sec. 3] identified three types of proper rotations, namely, Type I, covering matrices of the form R=I3; Type II, covering matrices of the form R=2aa′-I3, where ∥a∥=1; Type III, covering matrices of the form R=I3+fa(Ta+Ta2), where fa=2/(∥a∥2+1). Moreover, it was pointed out in [6] that each proper rotation can be attributed to one of these types. Direct calculations show that rotations of Type I are obtained from the representation (1.5) by taking α=0, β=1, γ=0; Type II are obtained from the representation (1.5) by taking α=0, β=-1, γ=2, and ∥a∥=1; Type III are obtained from the representation (1.5) by taking α=fa, β=1-fa∥a∥2, γ=fa, where fa=2/(∥a∥2+1). Combining these observations with Corollary 2.6 leads to the conclusion that rotations of Type I have eigenvalues μ1=1, μ2=1, μ3=1; Type II have eigenvalues μ1=1, μ2=-1, μ3=-1; Type III have eigenvalues μ1=1, μ2=1-fa∥a∥(∥a∥-i), μ3=1-fa∥a∥(∥a∥+i). Furthermore, from Theorem 2.11 we obtain the following characterizations of the eigenspaces.Corollary 2.12.
LetR∈Υ be a proper rotation, and let  ℰ(μj),  j=1,2,3, be the eigenspaces of  R associated with its eigenvalues  μj. Then, (i)
ifR is of Type I, then  ℰ(μ1)=ℂ3,1,  ℰ(μ2)=ℂ3,1,  ℰ(μ3)=ℂ3,1,(ii)
ifR is of Type II, then  ℰ(μ1)=ℛ(a),  ℰ(μ2)=ℛ(Qa),  ℰ(μ3)=ℛ(Qa),(iii)
ifR is of Type III, then  ℰ(μ1)=ℛ(a),  ℰ(μ2)=ℛ(Q1),  ℰ(μ3)=ℛ(Q3), where  Qk=I3-Pk,  k=1,3, with  Pk as specified in  (2.3).



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*Source: 578352-2011-06-21.xml*