Ortho-Skew and Ortho-Sym Matrix Trigonometry 1

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1 Abstract Ortho-Skew and Ortho-Sym Matrix Trigonometry Daniele Mortari Department of Aerospace Engineering, Texas A&M University, College Station, TX This paper introduces some properties of two families of matrices: the Ortho-Skew, which are simultaneously Orthogonal and Skew-Hermitian, and the real Ortho-Sym matrices, which are Orthogonal and Symmetric. These relationships consist of closed-form compact expressions of trigonometric and hyperbolic functions that show that multiples of these matrices can be interpreted as angles. The analogies with trigonometric and hyperbolic functions, such as the periodicity of the trigonometric functions, are all shown. Additional expressions are derived for some other functions of matrices such as the logarithm, exponential, inverse, and power functions. All these relationships show that the Ortho-Skew and the Ortho-Sym matrices can be respectively considered as matrix extensions of the imaginary and the real units. Introduction The computation of functions of square matrices, real or complex, is an important topic in many engineering application fields and in applied mathematics. Evaluation of exponential, logarithm, powers, as well as trigonometric, hyperbolic and many other functions is often needed. Unfortunately, in general, this problem does not admit closedform solutions. However, as it will be shown in this paper, it is possible to develop Work previously presented as paper AAS of the John L. Junkins Astrodynamics Symposium, College Station, TX, May 3-4, 003. Associate Professor, Department of Aerospace Engineering, 70 H.R. Bright Building, Room 74A, Texas A&M University, 34 TAMU, College Station, Texas Tel. (979) , Fax (979) AIAA and AAS Member.

2 compact closed-form solutions for several different functions of two families of matrices: the Ortho-Skew and the Ortho-Sym matrices. There are several methods to evaluate functions of a matrix. In particular, the books [], [], [3], and [4] dedicate full chapters on functions of a matrix. The function of a matrix f(m) can be introduced and defined in different ways that are all equivalent [5]. For instance, there is the line integral definition[6] f(m) = πi Γ f(z) (zi M) dz () that is computed on a closed contour Γ that encircle the eigenvalues λ of M. As noted by [3], this definition is the matrix version of the Cauchy integral theorem. In fact, respectively writing f k,j and g k,j (z) for the (k, j) entries of f(m) and (zi M), Equation () can be written in scalar form f k,j = πi f(z) g k,j (z) dz () Γ Other existing methods to define f(m) use the Jordan decomposition M = W J W [7, 8], where W and J are respectively the eigenvector matrix and the Jordan normal form of M; or the Schur decomposition M = H T H, where H and T are respectively orthogonal and upper triangle matrices [9]. Using either decomposition, we can define f(m) as f(m) = W f(j) W and f(m) = H f(t ) H (3) In the Jordan decomposition, f(j) is a block diagonal matrix with upper triangle blocks f i (J) that are associated with the J i blocks of J accordingly with λ i 0 0 f i f () f () i f i! (m i ) i (m i )! 0 λ i 0 J i = 0 0 λ i 0 0 f i f () f i (m i ) i (m i )! f i (J) = f. 0 0 f i (m i 3) i (4) (m i 3)! λ. i f i where λ i is the eigenvalue of the m i m i block J i, f i :=f(λ i ), and f (k) i is the kth derivative of f(x) computed at λ i. Note also that when M is diagonalizable, the computation of W in Eq. (3) is unnecessary because of the Sylvester (or Lagrange-Sylvester) formula

3 [] that requires only the computation of the n eigenvalues of M f(m) = n f(λ j ) Z j where Z j = j= n i=,i j (λ j λ i ) (M λ ii) (5) where the eigenvalues of M (correctly counting multiplicities) are λ,..., λ n. The Jordan decomposition given in Eq. (3) is a specific case of a more general relationship relating two similar matrices, A and B. In this case, if A = C B C then f(a) = C f(b) C (6) It is to be noted that there are some computational difficulties with the Jordan decomposition approach, particularly when M is defective (non-diagonalizable) or has ill-conditioned eigenvectors [3]. For the Schur decomposition, once H and T have been evaluated, the computation of f(m) is reduced to the computation of f(t ), where T is upper triangular. Explicit expressions for f(t ) easily become very complicated. To overcome this problem, Parlett [9] has introduced a recursive method that has completely solved this problem. The Jordan and Schur decomposition methods are more general in the sense that an arbitrary function of a given square matrix can be computed using these algorithms. Another general method to define any f(m) that will extensively used throughout this paper comes from the power series representation of functions or, in other words, from the Taylor series. This implies that, if f(x) = a k x k then f(m) = a k M k (7) Equation (7) holds if the f(λ i ) is analytic in a neighborhood of 0 and the radius of convergence of f contains all the λ i, where λ i are the eigenvalues of M. In particular, f(x) = k! d k f dx k x k then f(m) = x=0 k! d k f dx k M k (8) x=0 The use of Eq. (8) clearly implies some additional numerical approximation work and for this reason its use is rather limited 3. However, this paper nevertheless shows that the Maclaurin series expansion yields a compact closed-form solutions for several 3 The rational Padé approximation [3] is commonly used to approximate the specific and important case of matrix exponentials. 3

4 matrix functions in the special case of two important matrix families: the Ortho-Skew and the Ortho-Sym matrices. In particular, for the case of trigonometric and hyperbolic matrix functions, the similarity with the scalar formulations is so close that one can justifiably speak of matrix trigonometry. The Ortho-Skew and Ortho-Sym Matrix Sets The Ortho-Skew matrices I C n n are simultaneously Orthogonal and Skew-Hermitian I I = I and I = I = I = I (9) Reference [0] has shown that the Ortho-Skew matrix set presents very interesting properties, some of which, are summarized in the following. The general expression of these matrices, whose eigenvalues are always ±i, is I = i p n c k c k i c k c k where c i c j = δ ij (0) k= k=p+ where p indicates the number of positive eigenvalues +i (0 p n). Note that, if p = n then I = i I, and if p = 0 then I = i I. Equation (9) then clearly implies that the powers of an Ortho-Skew matrix obey the following simple rule I k = ik + ( i) k I i ik ( i) k ( I = cos k π ) ( I + sin k π ) I () In general, an Ortho-Skew matrix is complex. However, in even-dimensional spaces, it is possible to build real I matrices as follows n/ [ I = P k I Pk T where P k = [ c k. c k ] and I = k= 0 0 is the simplectic matrix. Similarly the Ortho-Sym real matrices R R n n, which have been introduced in Ref. [], are simultaneously Orthogonal and Symmetric ] () R T R = I and R = R T = R = I (3) The powers of these matrices clearly obey the following simple rule R k = + ( )k I + + [ ] [ ] ( )k+ + cos(kπ) cos(kπ) R = I + R (4) 4

5 The general expression of these matrices, whose eigenvalues are always ±, is p n R = r k r T k r k r T k where r T i r j = δ ij (5) k= k=p+ where p (0 p n) is the number of positive eigenvalues +. Since the r k are orthogonal, if p = n, then R = I whereas if p = 0, then R = I. Equations () and (4) then allow us to give compact closed form expressions for certain infinite series expansions of the Ortho-Skew and the Ortho-Sym matrices in the following way: Given any function f : C C analytic in a disc of radius > centered at the origin, assume it s Maclaurin expansion is f(x)= c k x k. Clearly then, Eq. () implies f(i) = and Eq. (4) implies = = f(r) = = = c k I k = [ c k i k + ( i) k f(i) + f( i) c k R k = [ Exponential functions [ ] i k + ( i) k c k I i ik ( i) k I (6) ] [ ] i k ( i) k I i c k I (7) I i c k + ( ) k f() + f( ) Equation (8), with f(x) = e αx yields f(i) f( i) I (8) [ + ( ) k c k I + + ] ( )k+ R (9) ] [ ] + ( ) k+ I + c k R (0) I + f() f( ) R. () e Iα = I cos α + I sin α () which gives a matrix extension of the fundamental DeMoivre formula e i α = cos α + i sin α (3) 5

6 to the matrix field. In this light, we can see I as a matrix generalization of the imaginary unit (or Ortho-Skew matrix). Similarly, Eq. () allows us to express e Rα as follows e Rα = I cosh α + R sinh α (4) which yields a matrix extension of the fundamental formula e α = cosh α + sinh α (5) In this case, the real unit can be considered a Ortho-Sym matrix. Equations () and (4) yield the trigonometric identities I cos α = eiα + e Iα and I sin α = eiα e Iα (6) and the hyperbolic identities I cosh α = erα + e Rα and R sinh α = erα e Rα (7) Trigonometric and Hyperbolic Functions Equation (8), with f(x)=sin(αx) or f(x)=cos(αx) yields sin(αi) = I sinh α and cos(αi) = I cosh α (8) From Eq. (8), it is possible to evaluate tan(αi) = I tanh α and cot(αi) = I coth α (9) Similarly, for the Ortho-Sym matrices, we have the relationships sin(αr) = R sin α and cos(αr) = I cos α (30) which allows us to evaluate the tangent function tan(αr) = R tan α and cot(αr) = R cot α (3) In the same way, we can also derive the following hyperbolic functions sinh(αi) = I sin α and cosh(αi) = I cos α (3) 6

7 From this equation, we can also evaluate the following expressions tanh(αi) = I tan α and coth(αi) = I cot α (33) Similarly, the hyperbolic functions of Ortho-Sym matrices are sinh(αr) = R sinh α and cosh(αr) = I cosh α (34) which allows us to evaluate the hyperbolic tangent functions tanh(αr) = R tanh α and coth(αr) = R coth α (35) The effect of the cosine and the hyperbolic cosine applied to αi and αr is interesting, for the result in both cases is a diagonal matrix which depends on the constant α only. Put another way, the cosine and the hyperbolic cosine of I and R completely delete all the eigenvalue and eigenvector information contained in these matrices. It appears that the cosine functions are a sort of projection aligned with all the eigenvalues and eigenvectors of these two matrix families. In fact, for I I and R R, we can write that { cos(αi ) = cos(αi ) = cosh(αr ) = cosh(αr ) = I cosh α (36) cosh(αi ) = cosh(αi ) = cos(αr ) = cos(αr ) = I cos α From Eqs. (8) and (3), we can derive that { cos (αi ) + sin (αi ) = I cosh (αi ) sinh (αi ) = I while, from Eqs. (30) and (34), we obtain { cos (αr ) + sin (αr ) = I cosh (αr ) sinh (αr ) = I (I I ) (37) (R R ) (38) It is to be noted that the relationships given in Eqs. (37) and (38) are quite different from the identities { cos (M) + sin (M) = I cosh (M) sinh (39) (M) = I that, in turn, hold for just a single (but arbitrary) matrix M. Equations (8), (30), (3), and (34), allow us to derive the general power expressions for the sines when k is even { sin k (αi) = ( ) k/ I sinh k α sinh k (αi) = ( ) k/ I sin k α 7 { sin k (αr) = I sin k α sinh k (αr) = I sinh k α (40)

8 while when k is odd we have { sin k (αi) = ( ) (k )/ I sinh k α sin k (αr) = R sin k α { sinh k (αi) = ( ) (k )/ I sin k α sinh k (αr) = R sinh k α (4) as well as for the cosines { cos k (αi) = I cosh k α = cosh k (αr) cosh k (αi) = I cos k α = cos k (αr) (4) Inverse Trigonometric and Hyperbolic Functions As for the inverse functions, they can be derived straightforwardly from our preceding expressions. Let us, just for example, derive one of them. From the identity sin(βi) = I sinh β given in Eq. (8), we obtain sin [sin(βi)] = βi = sin (I sinh β). (43) Now α = sinh β β = sinh α, so Eq. (43) becomes sin (αi) = I sinh α. By analogous procedures, we derive the following inverse trigonometric functions sin (αi) = I sinh α tan (αi) = I tanh α cot (αi) = I coth α and sin (αr) = R sin α tan (αr) = R tan α cot (αr) = R cot α (44) while, for the inverse hyperbolic functions, we obtain sinh (αi) = I sin α tanh (αi) = I tan α coth (αi) = I cot α and sinh (αr) = R sinh α tanh (αr) = R tanh α coth (αr) = R coth α (45) The inverse trigonometric cosine functions have the expressions cos (αi) = π I I sinh α and cos (αr) = π I R sin α (46) derived from the trigonometric identity cos α = π sin α (47) and from Eq. (44). The expressions of the inverse hyperbolic functions are cosh (αi) = π I + I sinh α and cosh (αr) = i π I i R sin α (48) 8

9 that can be derived from the scalar hyperbolic identity Equations (46) and (48) allow us to write the identities cosh (iα) = i π + sinh α (49) cosh (αi) = I cos (αi) and cosh (αr) = i cos (αr) (50) Trigonometric and Hyperbolic Functions Periodicity An immediate consequence of our preceding formulae is that the Ortho-Skew and the Ortho-Sym matrices fully respect the symmetry and periodicity of the trigonometric and hyperbolic functions. For instance, we have cos( αi) = cos(αi), sin( αi) = sin(αi), tan( αi) = tan(αi), cot( αi) = cot(αi), and also cos( αr) = cos(αr), sin( αr) = sin(αr), tan( αr) = tan(αr), cot( αr) = cot(αr). As for the kπ, and the kπ function periodicity, simple analogues of Eqs. (8) and () easily yield the following formulae sin(αi) = sin(αi + kπi) sin(αi ± πi) = sin(αi) cos(αi) = cos(αi + kπi) cos(αi ± πi) = cos(αi) tan(αi) = tan(αi + kπi) cot(αi) = cot(αi + kπi) and and the obvious analogous formulae hold for shifts by Iπ/ sin(αr) = sin(αr + kπi) sin(αr ± πi) = sin(αr) cos(αr) = cos(αr + kπi) cos(αr ± πi) = cos(αr) tan(αr) = tan(αr + kπi) cot(αr) = cot(αr + kπi) (5) { cos(αi) = sin(αi + π/i) tan(αi) = cot( αi + π/i) and { cos(αr) = sin(αr + π/i) tan(αr) = cot( αr + π/i) (5) All the relationships given in Eqs. (5) and (5) can be demonstrated through spectral decomposition and series expansions. As an example, let us demonstrate the identity cos(αr + kπi) = cos(αr) = I cos α. Let R = CΛC T be the spectral decomposition of R. Then we can write αr + kπi = C(αΛ)C T + C(kπI)C T = C(αΛ + kπi)c T, while the series expansion of the cosine allows us to write [ ] ( ) n cos(αr + kπi) = C (αλ + kπi)n C T (53) (n)! n=0 9

10 First, we notice that (αλ + kπi) n are powers of diagonal matrices, that can be substituted by powers of their scalar diagonal elements. Secondly, Λ contains only elements λ = ± and, therefore, these scalars can be written in the form kπ ± α, which means that the series for the diagonal elements (the only nonzero elements) converges either to cos(kπ + α) or to cos(kπ α), both equal to cos α. Therefore, Eq. (53) can be re-written as cos(αr + kπi) = C I cos α C T = I cos α = cos(αr) (54) which, thanks to Eq. (36), allows us to write cos(αr + mπi) = cos(αr + nπi) = I cos α cosh(αi + mπi) = cosh(αi + nπi) = I cos α cos(αi + mπi) = cos(αi + nπi) = I cosh α cosh(αr + mπi) = cosh(αr + nπi) = I cosh α for any values of the integers m and n, and for R R, and I I. The demonstration of the other formulae follows similar paths. (55) Power Function As for the power series, it is always possible to write { (αi + βi) n = I + (n even) (αr + βi) n I 4 R = 3 I + For even values of n, the coefficients of Eq. (56) have the closed-form expressions n/ ( ) = n! β n ( ) j j α (n j)! (j)! β n/ = n! β n 3 = n! β n n/ n/ 4 = n! β n ( ) j (n j )! (j + )! (n j)! (j)! ( ) j α β (n j )! (j + )! ( ) j+ α β ( ) j+ α β (56) (57) 0

11 while, for odd values of n, the coefficients become (n )/ = n! β n (n odd) (n )/ = n! β n (n )/ 3 = n! β n (n )/ 4 = n! β n ( ) j (n j)! (j)! ( ) j α β ( ) j (n j )! (j + )! (n j)! (j)! ( ) j α β (n j )! (j + )! ( ) j+ α β ( ) j+ α β (58) Logarithmic Function The logarithmic functions ln(i + M) has the series expansion [] ln(i + M) = ( ) k M (k+) (k + ) (59) Therefore, we have ln(i + αi) = = I ( ) k α(k+) (k + ) + I ( ) k αk+ k + (60) The first (alternating) series in Eq. (60) converges to ln( + α ), while the second is the inverse trigonometric tangent series expansion. Therefore, we obtain the solution ln(i + αi) = I ln( + α ) The logarithmic series for the Ortho-Sym matrices + I tan α (α ±i) (6) ln(i + αr) = = I α (k+) (k + ) + R α k+ k + (6)

12 The first series converges to ln( α ), while the second coincides with the inverse hyperbolic tangent series expansion. Therefore, Eq. (6) becomes Inverse Function ln(i + αr) = I ln( α ) Applying the geometric series identity results in (I αi) = I + R tanh α (α ±) (63) x = x k to the Ortho-Skew matrices, ( ) k α k + I ( ) k α k+ = (I + αi) (64) + α (α ±i) while, for the Ortho-Sym matrices, it gives (α ±) (I αr) = I α k + R α k+ = (I + αr) (65) α Equations (64) and (65) allow us to write { (I + I)(I I) = (I I) (I + I) = I (I + i R)(I i R) = (I i R) (I + i R) = i R (66) which are Isomorphic Cayley Transforms since they map the I and the i R matrices onto themselves, respectively. The first of Eq. (66) can be easily understood since the Cayley function f(z) = z maps z = ±i into f(z) = i [3] which are, in turn, the + z eigenvalues of I. The second of Eq. (66) involves the pure imaginary matrix ir which, in turn, has eigenvalues λ = ±i only. This is the main reason that Cayley Transforms become isomorphic for this matrix. Conclusion This paper presents some interesting compact expressions of matrix functions of two families of matrices: the Ortho-Skew and the Ortho-Sym matrices. The Ortho-Skew matrices consist of complex matrices (that can be real in even-dimensional spaces),

13 which are simultaneously Unitary and Skew-Hermitian, while the Ortho-Sym matrices are real matrices that are Orthogonal and Symmetric. Based mainly on the simplicity of their powers, simple analytical expressions can be derived from the series expansions of many matrix functions for these two kind of matrices. In particular, the simplicity of the presented formulae and their analogies with the scalar trigonometric and hyperbolic functions clearly allow one to speak of matrix trigonometry of Ortho-Skew and Ortho-Sym matrices. Additionally, other matrix functions such as multiplicative inverse, powers, and the logarithmic functions are taken into consideration and compact general expressions have been found for them. Acknowledgements This paper is my personal gift to Dr John L. Junkins in occasion of his sixtieth orbit about the Sun. I want to thank John for sharing part of this most recent orbit with me, for his big friendship, for his modesty, and for being a living example that working and dreaming may coincide! I strongly feel in dept with my colleague Dr Maurice J. Rojas, for the generous help given in revising this manuscript and in making it more clear, compact, and simple. I also want to thank two anonymous referees for their helpful suggestions. References [] S. Barnett, Matrices: Methods and Applications, Clarendon Press, Oxford University Press, 99. [] F. R. Gantmacher, The Theory of Matrices, Chelsea Publishing Company, New York, N.Y., 990. [3] G. H. Golub and C. V. Loan, Matrix Computations, The Johns Hopkins Univeristy Press, Baltimore, MD, 983. [4] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 99. [5] R. F. Rinehart, The Equivalence of Definitions of a Matrix Function, American Math. Monthly, Vol. 6, pp

14 [6] N. Dunford and J. Schwartz, Linear Operators, Part I, Interscience, New York, N.Y., 958. [7] J. S. Frame, Matrix Functions and Applications, Part II, IEEE Spectrum, April 964, pp [8] J. S. Frame, Matrix Functions and Applications, Part IV, IEEE Spectrum, June 964, pp [9] B. N. Parlett, A Recurrance among the Elements of Functions of Triangular Matrices, Linear Algebra and its Applications, Vol. 4, 976, pp. 7. [0] D. Mortari, On the Rigid Rotation Concept in n Dimensional Spaces, Journal of the Astronautical Sciences, Vol. 49, No. 3, July September 00, pp [] A. K. Sanyal, Geometrical Transformations in Higher Dimensional Euclidean Spaces, Master s thesis, Department of Aerospace Engineering, Texas A&M University, College Station, TX, May 00. [] G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw Hill Book Company, Avenue of the Americas, New York, N.Y. 000, Second ed., 968. [3] D. Mortari, Conformal Mapping among Orthogonal, Symmetric and Skew Symmetric Matrices, Advances in the Astronautical Sciences, AAS Paper of the AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico, February 9 3,

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