Conformal Mapping among Orthogonal, Symmetric, and Skew-Symmetric Matrices

Transcription

1 AAS Conformal Mapping among Orthogonal, Symmetri, and Skew-Symmetri Matries Daniele Mortari Department of Aerospae Engineering, Texas A&M University, College Station, TX Abstrat This paper shows that Cayley Transforms, whih map Orthogonal and Skew- Symmetri matries, may be onsidered the extension to matrix field of the omplex onformal mapping funtion f 1 (z) = 1 z. Then, by using a set 1 + x of real matries whih are, simultaneously, Orthogonal and Symmetri (the Ortho Sym matries), it similarly shows how to extend two omplex onformal mapping funtions (namely, the f (z) = i z i + z, and the f 3(z) = 1 z 1 + z i - here alled the lokwise, and the ounter lokwise funtions), to matrix field. This extension onsists of some new one-to-one mapping relationships between Orthogonal and Symmetri, and between Symmetri and Skew-Symmetri matries. This new relationships omplete the piture of the one-to-one matrix mapping among Orthogonal, Symmetri, and Skew- Symmetri matries. Finally, this paper shows how to map among Orthogonal, Symmetri, and Skew-Symmetri matries, by means of a diret produt. Introdution Cayley Transforms are a beautiful and useful tool to perform one-to-one mapping between Orthogonal C and Skew-Symmetri Q real matries. Cayley Transforms onsists of two formally idential relationships that an be written as { C = (I Q)(I + Q) 1 = (I + Q) 1 (I Q) (Forward) Q = (I C)(I + C) 1 = (I + C) 1 (I C) (Inverse) (1) where I indiates the identity matrix, and C T C = I, and Q = Q T. Assoiate Professor of Aerospae Engineering, Department of Aerospae Engineering, 701 H.R. Bright Building, Room 741A, Texas A&M University, 3141 TAMU, College Station, Texas Tel. (979) , FAX (979) AIAA and AAS Member. 1

2 Several appliations of these equations an be found in attitude dynami and ontrol. In fat, if C represents the diretion osine matrix desribing a given attitude (or rigid rotation), then the mapped Q matrix ontains the three elements of the Gibbs vetor, as the off-diagonal elements, a minimum number parameters to desribe the attitude. Moreover, it is easy to derive the following relationships (See pp of Ref. 1), whih relate the angular veloity ω with the rate of hange of C and Q ω = (I + Q) 1 Q(I Q) 1 = ĊCT = 0 ω 3 ω ω 3 0 ω 1 ω ω 1 0 () Another very important appliation of Cayley Transforms is for real Symmetri S matries. In fat, any Symmetri matrix an be deomposed as S = CΛ S C T = (I Q)(I + Q) 1 Λ S (I Q) 1 (I + Q) (3) that is, a minimal parametrization to desribe Symmetri matries. Equations (1,,3) an be speialized to any n-dimensional Eulidean spae. Finally, Eqs. (1) are extensively applied in all the fields where matrix transformation plays a key role. An important aspet of the Cayley Transforms, easily to demonstrate, is the fat that C and Q share the same omplex eigenvetor matrix W. In fat, let CW = W Λ C be the spetral deomposition of C, then, sine W W = I, we an write Q = (I C)(I + C) 1 = (W IW W Λ C W )(W IW + W Λ C W ) 1 = = W (I Λ C )W W (I + Λ C ) 1 W = W (I Λ C )(I + Λ C ) 1 W = = W Λ Q W where Λ Q = (I Λ C )(I + Λ C ) 1, that in salar form beomes λ Q = 1 λ C 1 + λ C (4) whih shows that the eigenvalues of C and Q are related through a bilinear onformal transformation. Therefore, Cayley Transforms represent the extension to matrix field of the partiular omplex onformal mapping funtion f 1 (z) = 1 z. The mapping 1 + z rule of this funtion, here alled Cayley funtion f 1 (z), is shown in Fig. 1, where the presene of a mirror helps to understand the mapping geometry. The relationship between Cayley mapping and the omplex onformal funtion f 1 (z), arises the question whether it is possible to develop similar matrix mapping with different omplex onformal funtions. In partiular, we will look to develop the Symmetri matries are widely used in mehanis (as representing inertia and stress tensors, or as mass, stiffness, and damping matries), as well as in dynamis and ontrol (as representing ovariane matries in Kalman filter, gain matries in optimal ontrol, and as solution of the matrix Lyapunov and the Riati differential equations).

3 missing similar matrix mapping to omplete the mapping between the three most ommon real n n matries: the Orthogonal C, the Symmetri S, and the Skew- Symmetri Q matries, respetively. To this end, the problem to share the real eigenvetor matrix P of S with the omplex eigenvetor matrix W of C and Q, arises. Spetral deomposition properties of C, Q, and S are summarized in Table 1. Matrix Definition Eigenvalues Eigenvetors Orthogonal C T C = I Unit-Cirle Unitary Skew-Symmetri Q = Q T Imaginary Axis Unitary Symmetri S = S T Real Axis Orthogonal Table 1: Spetral deomposition properties for C, Q, and S matries. The eigenvetor problem will be solved thanks to the Nuleus matrix M, while the eigenvalue problem (Whih are the mapping funtions to map the eigenvalues) will be solved by introduing the Counter Clokwise f (z), and the Clokwise f 3 (z) omplex mapping funtions. However, prior to enter into these arguments, the Ortho-Skew, and the Ortho-Sym matries, will be introdued. The Ortho-Skew I, and the Ortho-Sym R matries The intersetion of the unit irle (eigenvalue lous of the Unitary matries) and the imaginary axis (eigenvalue lous of the Skew-Hermitian matries) yields to matries whih are, simultaneously, Unitary and Skew-Hermitian. These matries an be, therefore, defined by the relationships { I I = I, I = I } = II = I (5) The resulting matrix I is, therefore, Unitary and Skew-Hermitian matrix set, here alled improperly Ortho-Skew, whih has been introdued in Ref. 3, presents very interesting properties. The general expression of these matries, whose eigenvalues are only pure imaginary ±i, is I = i p w k w k i n k=p+1 w k w k (w i w j = δ ij ) (6) where the omplex diretions w k, whih form the unitary eigenvetor matrix W onstitute an Orthogonal set, and where p indiates the number of positive eigenvalues +i (0 p n). Note that, sine the w k are Orthogonal, then if p = n then I = i I, while if p = 0 then I = i I. 3

4 The Ortho-Skew-Hermitian matries are, in general, omplex. However, in even dimensional spaes, it is possible to build real I e matries as follows I e = n/ n/ W k I W k = w k w k that an be simply alled Ortho-Skew. i 0 0 i Based on Eq. (5), subsequent powers of I satisfy the following rule = +I (k = 1, 5, 9, ) I k = I (k =, 6, 10, ) = = I (k = 3, 7, 11, ) = +I (k = 4, 8, 1, ) w k w k (7) Analogously, the intersetion of the unit-radius irle (eigenvalue lous of C) with the real axis (eigenvalue lous of S) yields to the definition of the Ortho-Sym R matries, whih are, simultaneously, Orthogonal and Symmetri. The Ortho-Sym real matries R, whih have been introdued in Ref. 4, are defined by their own definitions (8) R T R = I and R = R T (9) Subsequent powers of these matries obey to the simple rule R k = { = R (k = odd) = I (k = even) (10) The general expression of R, whose eigenvalues are only ±1, is p R = p k p T k n p k p T k (p T i p j = δ ij ) (11) k=p+1 where the real diretions p k form an orthogonal set and where p indiates the number of positive eigenvalues +1 (0 p n). Sine the p k are orthogonal, then if p = n then R = I, while if p = 0 then R = I. These matries, in partiular the Ortho-Sym matries (that an be built assoiated with a given C or S), will allow us to omplete the one-to-one mapping between Orthogonal, Skew-Symmetri, and Symmetri matries. The Nuleus Matrix The nuleus matrix is the omplex matrix 1 i M = 1 i (1) Also the omplex I matries will be here improperly alled Ortho-Skew. 4

5 This unitary matrix (MM = I ) allows to extrat the information of the plane, provided by a omplex-onjugate eigenvetor pair W k = ˆp 1k + iˆp k. ˆp 1k iˆp k = ŵ k. ŵ k, into a real plane defined by a real eigenvetor pair P k = ˆp 1k. ˆp k. In fat 1 i W k M = ˆp 1k + iˆp k. ˆp 1k iˆp k = ˆp 1 i 1k. ˆp k = P k (13) This property allows us to map a omplex unitary eigenvetor matrix W into an assoiated real Orthogonal matrix P. In fat, M M 0 P = P 1. P. p = W 1. W. p.... = W D(M) (14) where D(M) is a diagonal-blok matrix ontaining, as a diagonal elements, M matries assoiated with the n/ omplex eigenvalue pairs, and a one assoiated to the real eigenvalue (if n is odd, only). The nuleus matrix M allows us to build Ortho-Skew and Ortho-Sym matries, I(W ) and R(W ), whih are assoiated with a given unitary eigenvetor matrix W. In fat, for a even dimensional spae, we may write that where and where I(W ) = R(W ) = I = R = n/ n/ +i 0 0 i n/ W k I W k = P k R P T k = and n/ and P k M I M P T k = n/ W k M R M W k = n/ I (R) = M I M = R (I) = M R M = P k I (R) P T k W k R (I) W k The nuleus matrix M has many important properties. First of all, it allows to build all the three Pauli Spin Matries (See pp of Ref. ) σ 1 = = M M T σ 3 = (15) (16) (17) = M T M (18) and σ = M M M T M T M M T = 5 0 i i 0 (19)

6 whih are used in the omplex Cayley-Klein matrix to parameterize rigid rotation, and in quantum mehanis where they represent the angular-momentum matries for spin ( h). The seond property (really unusual), is that M as well as M T and M, all present a power periodiity of 4 whih allows us to write M 4 = (M T ) 4 = (M ) 4 = I (0) M 4l+k = (M T ) 4l+k = (M ) 4l+k = M k = (M T ) k = (M ) k (1) where l and k represent any integer. Conformal Complex Transformations A onformal mapping, also alled a onformal map, onformal transformation, anglepreserving transformation, or biholomorphi map, is a transformation w = f(z) that preserves loal angles. Let us onsider the following interesting omplex onformal mapping funtions (See pp of Ref. ), together with their inverses w = f 1 (z) = 1 z z = f 1 (w) = 1 w 1 + z 1 + w w = f (z) = i z i + z w = f 4 (z) = i z 1 + z z = f 3 (w) = 1 w 1 + w i z = f 4 (w) = i w 1 + w () w = f 5 (z) = 1 z i + z z = f 6 (w) = 1 iw 1 + w Table 1 summarizes the mapping results for these funtions when applied to the real axis (z = ±Φ), the imaginary axis (z = ±iφ), and the unit-irle (z = e ±i Φ ). From Real axis From Imaginary axis From Unit-Cirle f 1 (z) Real Unit-Cirle Imaginary f (z) Unit-Cirle Real Imaginary f 3 (z) Imaginary Unit-Cirle Real f 4 (z) I = R + 1 Cirle at 1, +1/ I = R f 5 (z) Cirle at 1, 1/ I = R 1 I = R f 6 (z) I = R 1 Cirle at +1, 1/ I = R Table 1: Mapping summary of w = R + i I = f(z). 6

7 It is easy to demonstrate that the first four of these omplex mapping funtions obey to the following general rules { { z = f1 (f 1 (z)) z = f (f and (f (z))) (3) z = f 4 (f 4 (z)) z = f 3 (f 3 (f 3 (z))) In the following setions, how to extend the first three omplex onformal funtions, f 1 (z), f (z), and f 3 (z), to matrix mapping, is shown. The analysis for the remaining three funtions, f 4 (z), f 5 (z), and f 6 (z), whih are muh more ompliated, will be the subjet of a future work. The Cayley mapping funtion f 1 (z) = 1 z 1 + z As already stated, Cayley Transforms, given in Eq. (1), an be seen as the extension to matrix field of this mapping funtion. In fat, the eigenvalues of C and Q are related through the omplex trigonometri identity ( ) Φ ( ) Φ i tan = 1 e±iφ e ±iφ = 1 + e ±iφ i tan ( ) (4) Φ i tan The geometry assoiated with this trigonometri identity or with the Cayley mapping funtion f 1 (z), is shown in Fig. 1, where the point 1, 0 works like a mapping refleting point. Φ / Im i tan Φ Φ e iφ Figure 1: f 1 (z) mapping rules Re Q Im U S S 1 Cayley funtion Re Figure : f 1 (z) = 1 z 1 + z funtion Figure summarizes the mapping properties of the f 1 (z) funtion. In partiular, f 1 (z) maps the real axis onto the real axis. This property an also be seen as the 7

8 trigonometri identities os Φ = ( ) Φ 1 tan ( ) Φ ( ) tan Φ 1 + tan = 1 os Φ 1 + os Φ (5) whose equivalent matrix mapping has the form { S1 = (I S )(I + S ) 1 = (I + S ) 1 (I S ) S = (I S 1 )(I + S 1 ) 1 = (I + S 1 ) 1 (I S 1 ) (6) where S 1 and S are two Symmetri matries. These two matries have the same ( ) Φ eigenvetor matrix, and whose eigenvalues are λ k = os Φ k, and tan, respe- tively. Trigonometri/Matrix relationships It is well known that any real square matrix an be deomposed as a sum of a Skew- Symmetri and a Symmetri matries. Speified to an orthogonal matrix C, this deomposition is as follows ( ) ( ) C + C T C C T C = + = S + A (7) where S and A have the eigenvalue pairs os Φ k, and ±i sin Φ k, respetively. ( ) Therefore, it is possible to apply the first of Eq. (6) with S = S =. The C + C T transformed matrix S 1, as it easy to demonstrate, an be set as S = QQ T, whih satisfies the two requested properties of having the same eigenvetor matrix of C, and whose eigenvalue pairs are obtained as Therefore, Eq. (6) beomes ( ) ( ) ( ) Φ Φ Φ i tan ±i tan = tan (8) { C + C T = (I QQ T )(I + QQ T ) 1 = (I + QQ T ) 1 (I QQ T ) QQ T = (I C C T )(I + C + C T ) 1 = (I + C + C T ) 1 (I C C T ) (9) Analogously, the trigonometri property ( ) Φ i tan ±i sin Φ = ( ) (30) Φ 1 + tan 8

9 allows us to write the assoiated matrix identity A = C C T = Q (I + QQ T ) 1 = (I + QQ T ) 1 Q (31) (the sign omes from the fat that, assoiated with Q we have the eigenvalues ( ) Φ i tan, while assoiated with A, we have the eigenvalues ±i sin Φ). Analogously, the definition of the tangent trigonometri funtion, whih an be written as ( ) Φ ±i sin Φ i tan ±i tan Φ = os Φ = ( ) (3) Φ 1 tan allows us to introdue a Skew-Symmetri matrix T that an be seen as a tangent matrix T = (C C T )(C + C T ) 1 = (C + C T ) 1 (C C T ) (33) that an also be expressed or deomposed as T = Q (I QQ T ) 1 = (I QQ T ) 1 Q (34) Analogously, the expression of the tangent half-angle ( ) Φ ±i sin Φ i tan = 1 + os Φ = 1 os Φ ±i sin Φ allows us to introdue the matrix identities { Q = (C C T )(I + C + C T ) 1 = (I + C + C T ) 1 (C C T ) Q = (I C C T )(C C T ) 1 = (C C T ) 1 (I C C T ) (35) (36) Equations (9), (31), (33), (34), and (36) and the assoiated trigonometri relationships given in Eqs. (8), (30), (3), and (35), are, therefore, stritly onneted. Note that, some of the above relationships represent a different way to write the Cayley Transforms. It is lear that matrix properties are substantially more omplex than trigonometri properties. However, the above relationships allows us to see the Orthogonal, Symmetri, and Skew-Symmetri matries as illuminated with an additional dimmer light. This small additional knowledge will ertainly help to understand the nature, and to develop the Matri Trigonometry, of whih the known one (the salar), only onstitutes a small subset. It is easy to demonstrate that the Ortho-Sym matrix R represents the key matrix to map Symmetri onto Skew-Symmetri matries, and vieversa. In fat, the relationship S = R(W ) A = A R(W ) A = R(W ) S = S R(W ) (37) where R(W ) means a Ortho-Sym matrix R whih is built with the eigenvetor matrix W of A, while the eigenvetor P = W D(M) of S is obtained by Eq. (14). 9

10 The Counter Clokwise funtion f (z) = i z i + z = 1 + iz 1 iz To extend the appliation of f (z) to matrix field (see Fig. 3 for its mapping summary), it is possible to demonstrate that, for the Symmetri matries S, the extension beomes { C = (I + RS)(I RS) 1 = (I RS) 1 (I + RS) C T = (I + RS) 1 (I RS) = (I RS)(I + RS) 1 (38) where R = R(W ) is the Ortho-Sym matrix assoiated with the eigenvetor matrix of C. The previous equation an be written also in the form { C = (R S)(R + S) 1 = (R S) 1 (R + S) C T = (R + S)(R S) 1 = (R + S) 1 (39) (R S) these equations implies the trigonometri identities ( ) Φ i tan ( ) = os Φ ± i sin Φ (40) Φ i ± tan Just for example, let us to demonstrate Eq. (39). If, as it happens in Cayley Transforms (that is, the formal struture of the eigenvalue mapping is idential to the matrix mapping), then we need to look for an unknown matrix X suh that C = (X S) (X + S) 1 is Orthogonal. From this equation we an easily derive that the unknown matrix X has the expression X = (I C) 1 (I + C) S. ( ) Φ Setting os Φ, s sin Φ, and t tan, we an write the expression for X in the -D ase 1 1 s 1 + s X s 1 s 1 + t 0 0 t = this matrix, whih is independent from Φ, is Symmetri, and Orthogonal, with eigenvalues ±1. This implies the onlusion that the unknown matrix X is the Ortho-Sym matrix R. Demonstration of Eq. (38) may be done on the same way. The f (z) mapping funtion, here named Counter Clokwise (beause it moves from real axis, to imaginary axis, to unit irle, and bak to real axis), annot be applied to Orthogonal and to Skew-Symmetri matries beause f (z) f (z ). In partiular, this mapping funtion satisfies f (e i Φ )f (e i Φ ) = 1 and f (i Φ)f ( i Φ) = +1 (41) 10

11 Im Counter Clokwise U funtion Im U Clokwise funtion Q Q Q U Q U Re Re S S S S Figure 3: f (z) = i z i + z = 1 + iz 1 iz funtion Figure 4: f 3 (z) = 1 z 1 + z i funtion The Clokwise funtion f 3 (z) = 1 z 1 + z i The mapping funtion f 3 (z) (see Fig. 4 for its mapping summary), an be used to find a way to map Orthogonal to Symmetri. The resulting mapping is S = (I C)(I + C) 1 R = (I + C) 1 (I C)R (4) whose demonstration is easily derived from Eq. (66) whih, in turn, an be seen as the a way to map Skew-Symmetri to Symmetri and vieversa S = QR = RQ and Q = SR = RS (43) The f 3 (z) mapping funtion, here named Clokwise (beause it moves from real axis, to unit-irle, to imaginary axis, and bak to real axis), annot be applied to Symmetri matries beause f 3 (Φ) f 3 ( Φ), and annot be applied to Skew-Symmetri matries beause f 3 (i Φ) f 3 ( i Φ). In partiular, this mapping funtion satisfies f 3 (Φ)f 3 ( Φ) = 1 and f 3 (i Φ)f 3 ( i Φ) = 1 (44) Mapping Summary All the above relationship are summarized below: From Orthogonal to Skew-Symmetri: Q = (I C)(I + C) 1 = (I + C) 1 (I C) (45) 11

12 From Skew-Symmetri to Orthogonal: C = (I Q)(I + Q) 1 = (I + Q) 1 (I Q) (46) From Orthogonal to Symmetri: S = (I C)(I + C) 1 R = (I + C) 1 (I C)R (47) From Symmetri to Orthogonal: C = (R S)(R + S) 1 = (R S) 1 (R + S) (48) From Skew-Symmetri to Symmetri: S = QR = RQR (49) From Symmetri to Skew-Symmetri: Q = SR = RS (50) The above set of equations onstitute the omplete matrix mapping relationships among Orthogonal, Skew-Symmetri, and Symmetri matries. From this set of equations it is possible to demonstrate the following R C R = C T R Q R = Q = Q T R S R = S and I C I = C I Q I = Q = Q T I S I = S (51) whih yield to the relationships R = i W D(I ) W and I = i W D(R ) W (5) where D(I ), and D(R ) are the n n diagonal matries obtained with the blok elements I, and R, respetively. Diret Produt Mapping The purpose of this setion is to find the relationships allowing one-to-one mapping among Orthogonal, Skew-Symmetri, and Symmetri matries, by a diret produt. This implies, for instane, to find out whih is the matrix G suh that S = G C maps a given orthogonal matrix C onto a symmetri matrix S, whih has assigned eigenvalues. Here again, the problem to share the eigenvetors, is solved using the Nuleus matrix. 1

13 In fat, setting k = os Φ k and s k = sin Φ k, the spetral deomposition of a given Orthogonal matrix C an be written as C = whih an also be written as W k Λ k W k + p pt where Λ k = k + is k 0 0 k is k (53) where n p C = W k MM Λ k MM W k + p pt = W k M = P k and M Λ k M = os Φ k sin Φ k P k C k P T k + p p T (54) sin Φ k = C os Φ k (55) k Therefore, the spetral deomposition of real Orthogonal matries an also be written in the full real form C = P k C k P T k + p p T where C k = Similarly, Skew-Symmetri matries an be expanded as A = P k A k P T k where A k = while for the Symmetri matries we have S = P k S k P T k where S k = Thanks to the eigenvetor orthogonality k s k 0 a k a k 0 rk 0 0 r k s k k (56) (57) (58) P T i P j = 0, i j; P T i P I ; P T i p = 0,1 ; p T p = 1. (59) the expressions provided by Eqs. (56-58) allows us to obtain ompat formulations for subsequent produts of Orthogonal, Symmetri, and Skew-Symmetri matries. For instane, using these expansions, the produt between an Orthogonal and a Symmetri matrix beomes C S = np i=1 P i C i P T i n p + p p T P j S j P T j (60) j=1 where all the terms with i j vanish beause of Eq. (59). Thus, C S = P k C k S k P T k = 13 P k F k P T k (61)

14 where F k = C k S k. This result provides us with the tool to develop general matrix mapping by means of a diret produt. In fat, the G matrix that maps the matrix B onto the final matrix F = G B an be built as follows G = F B 1 = P k G k P T k where G k = F k B 1 k (6) This proedure holds provided that the matries B k are not singular. The inverse mapping transformation (from F onto B) is simply G 1 = B F 1 = P k G 1 k P k T where G 1 k = B k Fk 1 (63) that holds if the matries F k are not singular. From the above, the following produt properties A S = S A C S = S C T C A = A C = { C S A = C A S = A C S A S C = S A C = S C A (64) are easily obtained. Speifially to our end, omitting the k index, setting s = sin Φ and = os Φ, and indiating with G BF and G k the matries used to map the B matrix onto the F matrix, we deal with the following nine ases: From Symmetri onto Symmetri G SS = S f S 1 rf 0 0 r f ri 0 0 r i 1 = r f r i I (65) From Skew-Symmetri onto Symmetri G AS = S f A 1 r 0 0 r 0 a a 0 1 = r a R (66) From Orthogonal onto Symmetri G US = S f C 1 r 0 0 r s s 1 = r s s (67) 14

15 From Symmetri onto Skew-Symmetri G SA = A f S 1 0 a a 0 r 0 0 r 1 = a r R (68) From Skew-Symmetri onto Skew-Symmetri G AA = A f A 1 0 ai a i 0 0 a f a f 0 1 = a i a f I (69) From Orthogonal onto Skew-Symmetri G UA = A f C 1 0 a a 0 s s 1 = a s s (70) From Symmetri onto Orthogonal G SU = C f S 1 s s r 0 0 r 1 = 1 r s s (71) From Skew-Symmetri onto Orthogonal G AU = C f A 1 s s 0 a a 0 1 = 1 a s s (7) From Orthogonal onto Orthogonal G UU = C f C 1 1 s 1 s 1 1 where 3 = os(φ 1 Φ ), and s 3 = sin(φ 1 Φ ) s s 1 = 3 s 3 s 3 3 (73) 15

16 Example The above equations, from Eq. (65) through Eq. (73), an be used, for instane, to keep the information of rotation (ontained in C), into Symmetri, Skew-Symmetri forms. An an example, let us to find the matrix mapping an Orthogonal matrix C onto Symmetri and Skew-Symmetri forms, whose eigenvalues are expressed as ( ) ( ) Φk tan Φk. S is obtained using Eq. (67), with r k = tan 4 4 ( ) Φk os Φ G k = G US = tan k sin Φ k (74) 4 sin Φ k os Φ k thus, the matrix m T US = P k G k Pk T (75) is suh that the matrix S = T US C (76) ( ) Φk is Symmetri and has eigenvalues ± tan. 4 Analogously, to obtain the Skew-Symmetri matrix, we should apply Eq. (70), with ( ) Φk a k = tan 4 ( ) Φk sin Φ G k = G UA = tan k os Φ k (77) 4 os Φ k sin Φ k thus, the matrix m T UA = P k G k Pk T (78) is suh that the matrix A = T UA C (79) ( ) Φk is Skew-Symmetri and has eigenvalues ±i tan. 4 Conlusion This paper shows that Cayley Transforms an be onsidered the extension to the matrix field of the omplex onformal mapping funtion f 1 (z). This results has been extended to similar matrix transforms whih map Orthogonal from/onto Symmetri, and Symmetri from/onto Skew-Symmetri matries. This has been ahieved Note that there is an expliit referene to the Modified Rodriguez Parameters (MRP), a minimum non-singular parameter set to desribe Orientation 16

17 using a set of real matries whih are simultaneously Orthogonal, and Symmetri (the Ortho Sym matries), and using two different omplex onformal mapping funtions (namely, the f (z), and the f 3 (z), that are here alled lokwise, and ounter lokwise funtions). Moreover, the lose relationship between some trigonometri identities and some matrix mappings is shown. This paper ends with a desription of a matrix mapping tehnique (always mapping Orthogonal, Symmetri, and Skew-Symmetri matries) that is obtained by diret matrix produt. Aknowledgment I would warmly like to thank Dr John L. Junkins and all the olleagues of the Aerospae Department of Texas A&M University for diretly and indiretly enouraging me to keep going with this researh. Referenes 1 J.L. Junkins and Y. Kim. Introdution to Dynamis and Control of Flexible Strutures. Amerian Institute of Aeronautis and Astronautis, AIAA Eduation Series, Washington, D.C., G.A. Korn and T.M. Korn. Mathematial Handbook for Sientists and Engineers. MGraw-Hill Book Company, 11 Avenue of the Amerias, New York, N.Y. 1000, Seond edition, D. Mortari. On the Rigid Rotation Conept in n-dimensional Spaes. The Journal of the Astronautial Sienes, 49(3):401 40, July-September A.K. Sanyal. Geometrial Transformations in Higher Dimensional Eulidean Spaes. Master s thesis, Department of Aerospae Engineering, Texas A&M University, College Station, TX, May