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Exponentials of Real Skew-Symmetric Matrices in Terms of Their Eigenvalues

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Exponentials of Real Skew-Symmetric Matrices in Terms of Their Eigenvalues EXPONENTIALS OF REAL SKEW-SYMMETRIC MATRICES IN TERMS OF THEIR EIGENVALUES A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Diego Gerardo Andree´ Avalos Galvez ´ 2018 SIGNATURE PAGE THESIS: EXPONENTIALS OF REAL SKEW-SYMMETRIC MATRICES IN TERMS OF THEIR EIGENVALUES AUTHOR: Diego Gerardo Andree´ Avalos Galv´ ez DATE SUBMITTED: Spring 2018 Department of Mathematics and Statistics Dr. Hubertus von Bremen Thesis Committee Chair Mathematics & Statistics Dr. Randall Swift Mathematics & Statistics Dr. Jennifer Switkes Mathematics & Statistics ii ACKNOWLEDGMENTS I would like to thank every math instructor I have had because they all have kindly shared their mathematical knowledge with me, put up with my carefree attitude, and helped shape the person that I am now. To my math primary and secondary school teachers, my community college instructors, and my graduate school professors, sincerely, thank you all. Specially Dr. von Bremen, who stands out for being not only a grad thesis adviser, but a life adviser, a motivator, and a true human being. Furthermore, I would like to thank Dr. Swift and Dr. Switkes for kindly accepting the invitation of being part of my thesis committee on such a short notice, and for their willingness to share their expertise in my thesis. I also would like to thank all my family in Peru, who, even when I lose contact with them for long periods of time, have always been proud of me: the first mathematician of the family. Most importantly, I want to express my deepest gratitude and admiration to my very first mentor: my mother, who never stopped believing in me and is always pushing me to achieve greater things, even during my most stubborn moments. I am truly the luckiest son in the world. iii ABSTRACT The eigenvalues of an n × n real nonzero skew-symmetric matrix S are purely imaginary or zero. Let the list of distinct purely imaginary eigenvalues of S be ±q1i;:::;±qpi such S that each q j > 0. We algebraically demonstrate that the exponential e can be expressed n−1 in terms of the powers In;S;:::;S , where the coefficients are in terms of the distinct S values q j, by using the method by Gallier and Xu [2]. Furthermore, the formulas of e (in terms of the q js) depend solely on the number of distinct eigenpairs ±q ji of S and whether zero is an eigenvalue of S, but are independent of their algebraic multiplicities. Only the formulas of the q js (in terms of the entries of S) depend on the multiplicities of the q jis in the characteristic polynomial of S. This allows us to determine that if n is even, S n−1 then e has n − 1 different cases, and 2 cases if n is odd. In this thesis, we calculate all the closed form formulas of eS for 2 ≤ n ≤ 9 because we can obtain the eigenvalues of S in terms of its entries up to the case n = 9 using the linear, quadratic, cubic, and quartic formulas. Nevertheless, the theory allows us to calculate the closed formula of eS for any arbitrary n assuming the eigenvalues of S are known. Lastly, we implement the formulas obtained in this thesis on our Matlab function skewexpm and compare the orthogonality errors using our formulas on randomly generated skew-symmetric matrices to those obtained by applying Matlab’s expm. It turns out that our formulas give a smaller error than expm for over 97% of the time up to size n = 5, over 92% of the time up to size n = 7, and for over 60% of the time for sizes n = 8 and 9 (see Table 4.1). Finally, if we allow the entries of a skew-symmetric matrix to range from −1015 to 1015, we can rely that our closed formulas will have a far better and acceptable error than expm, as our example with n = 9 illustrates in Figure 5.9. iv Contents 1 Introduction 1 2 Preliminary Concepts 5 3 The Theory 9 4 Computing the Formulas of eS 21 4.1 S 2 M1 . 21 4.2 S 2 M2 . 22 4.3 S 2 M3 . 22 4.4 S 2 M4 . 22 4.5 S 2 M5 . 25 4.6 S 2 M6 . 26 4.7 S 2 M7 . 30 4.8 S 2 M8 . 32 4.9 S 2 M9 . 40 4.10 Special Cases for S 2 Mn, n ≥ 10 . 46 5 Computer Results and Conclusions 48 v Bibiliography 61 A Roots of Polynomials 63 A.1 The Linear Case and the Quadratic Formula . 63 A.2 The Cubic Formula . 63 A.3 The Quartic Formula . 64 B Matlab Codes 65 B.1 The Skewexpm Function . 65 B.2 The Skewsymgenerator Function . 75 vi List of Tables 5.1 Expected Performance Rate of skewexpm against expm . 57 vii List of Figures 5.1 Case 4.2 (n = 2) Closed Formula Error vs. Matlab’s expm . 49 5.2 Case 4.3 (n = 3) Closed Formula Error vs. Matlab’s expm . 50 5.3 Case 4.4.3 (n = 4) Closed Formula Error vs. Matlab’s expm . 51 5.4 Case 4.5.2 (n = 5) Closed Formula Error vs. Matlab’s expm . 52 5.5 Case 4.6.5 (n = 6) Closed Formula Error vs. Matlab’s expm . 53 5.6 Case 4.7.3 (n = 7) Closed Formula Error vs. Matlab’s expm . 54 5.7 Case 4.8.7 (n = 8) Closed Formula Error vs. Matlab’s expm . 55 5.8 Case 4.9.4 (n = 9) Closed Formula Error vs. Matlab’s expm . 56 5.9 Case 4.9.4 (n = 9) Closed Formula Error vs. Matlab’s expm with Matri­ ces of Large Entries . 58 viii Chapter 1 Introduction The problem of numerically computing the exponential of a matrix A, where A is square and complex-valued, is difficult if we seek to have little error, and the method of choice to attempt to control this error usually depends on the properties of the matrix A. To better understand the problem we are facing, note that Moler and Van Loan [7] refer to all our existing methods of computing eA as dubious. For any general square matrix A, Moler and Van Loan summarize five classes of methods to compute its matrix exponen­ tial: series, ODE, polynomial, matrix decomposition, and splitting methods. Higham [5] studies two series methods, which are often used in computer algorithms to compute exponential of matrices (e.g., Matlab’s expm, and Mathematica’s MatrixExp). The two methods used to numerically compute the exponential of a matrix are scale and square, and Pade´ approximations. The algorithm consists of choosing a positive integer s so that 1 X jj 2s Ajj is appropriately bounded, and then use an [m=m] Pade´ approximant of e , namely, −1 a rational function R(X) = Q(X) P(X), where Q(X) and P(X) are two polynomials of degree m such that Q(X) = P(−X). Finally, the matrix exponential is approximated to A 1 2s s e ≈ [R( 2s A)] , and the power of 2 is dealt with by squaring repeatedly. However, er­ 1 ror problems may arise when using this method. For example, Golub and Van Loan [3] explain that some matrices may exponentially grow before decaying during the squaring 1 process of R( 2s A). Furthermore, Al-Mohy and Higham [1] himself address that if the integer s is not chosen carefully and ends up being too large, a phenomenon called over- scaling can cause a significant loss of accuracy. For instance, they present a new version of the scale and square method, which imposes more strict error bounds by choosing a minimal integer s, and m ≤ 13. This improved method is how matrix exponentials are computed in the newer versions of Matlab. In this thesis, we present yet another dubious method to calculate the matrix expo­ nential of one heavily-studied class of matrices: real skew-symmetric matrices S, by obtaining closed formulas of eS in terms of its eigenvalues and powers of S, and then compare our method with Matlab’s expm. One application where accurate computa­ tions of exponentials of skew-symmetric matrices are required is when computing the Lyapunov characteristic exponent of continuous dynamical systems using the eS-method described by von Bremen [11], where he uses real skew-symmetric matrices to charac­ terize orthogonal matrices Q in the SO(n) group with fewer entries than n2. In his paper, von Bremen presents closed formulas of eS for S of size 4 and 5 by using functions of matrices. Piamonte [9] obtains similar formulas to the ones contained in this paper up to size 7 by also using function of matrices, but with some missing cases and one overlap­ ping case. Moreover, Politi [10] and Oshinuga [8] independently calculated the formulas of eS when S is of size 4. We demonstrate that such formulas of eS can be calculated for any real skew-symmetric matrix S of arbitrary size n, as long as the eigenvalues are known, and to find the closed form formulas, one only needs to solve a linear system of equations of matrices. The method we use is from Gallier and Xu [2], who show that eS has a Rodrigues-like formula (see Section 4.3 to view Rodrigues’ formula), which 2 depends on the number of distinct nonzero eigenvalues of S. We show that the nonzero eigenvalues of a real skew-symmetric matrix are purely imaginary, so the characteristic polynomial P(X) of S has a coefficient of zero for each odd power of X when S has even size, or a zero for each even power of X if S has odd size.
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