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The Exact Methods to Compute the Matrix Exponential

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The Exact Methods to Compute the Matrix Exponential IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 1 Ver. IV (Jan. - Feb. 2016), PP 72-86 www.iosrjournals.org The Exact Methods to Compute The Matrix Exponential Mohammed Abdullah Salman1, V. C. Borkar2 1(Department of Mathematics& statistics Yashwant Mahavidyalaya, Swami Ramamnand Teerth Marthwada University, Nanded, India). 2(Department of Mathematics & Statistics, Yeshwant Mahavidyalaya, Swami Ramamnand Teerth Marthwada University, Nanded, India). Abstract: in this paper we present several kinds of methods that allow us to compute the exponential matrix e tA exactly. These methods include calculating eigenvalues and Laplace transforms are well known, and are mentioned here for completeness. Other method, not well known is mentioned in the literature, that don’t including the calculation of eigenvectors, and which provide general formulas applicable to any matrix. Keywords: Exponential matrix, functions of matrix, Lagrange-Sylvester interpolation, Putzer Spectral formula, Laplace transform, Commuting Matrix, Non-commuting Matrix. I. Introduction The exponential matrix is a very useful tool on solving linear systems of first order. It provides a formula for closed solutions, with the help of this can be analyzed controllability and ob servability of a linear system [1]. There are several methods for calculating the matrix exponential, neither computationally efficient [7,8,9,10]. However, from a theoretical point of view it is important to know properties of this matrix function. Formulas involving the calculation of generalized Laplace transform and eigenvectors have been used in a large amount of text books, and for this reason, in this work is to provide alternative methods, not well known, friendly didactic. There are other methods [4] of at s interesting but not mentioned in the list of cases because of its practicality in implementation. Eight cases or develop methods to calculate cases because of its practicality in implementation. Eight cases or develop methods to calculate the matrix exponential. Provide examples of how to apply the lesser-known methods in specific cases, and for the most known cases the respective bibliography cited. II. Definitions And Results The exponential of an n n complex matrix A denoted bye tA defined by ( At ) k ( At ) 2 ( At ) n 1 (t ) e At I At ...... .......... k ! 2! ( n 1)! k 0 To set the convergence of this series, we define firstly the frobenius norm of a matrix of size m n as follow 1 m n 2 2 A a F ij i 1 j 1 If A (:, j ) denotes the j-th column of A, and A (i, :) the ith row, it is easy to see that is satisfy 1 1 n 2 m 2 2 2 A A (:, j ) A (i. :) F 2 j 1 2 i 1 We will use this standard for convenience, because in a finite dimension vector space all norms are equivalent. An important property is to know how to use narrows the Frobenius norm of a matrix product. Given the m p p n matrices A and B then the product of themC ij AB , with entries ij A (i, :) B ( j, :) . If A had complex entries, we obtain conjugate C ij applied to row . Recall the Cauchy-Schwarz inequality 2 2 C ij A (i, :) B ( j, :) then we have : m n 2 m n 2 m 2 n 2 2 2 2 2 AB C A (i, :) B (: , j) A (i, :) B(:, j) A B . F ij 2 F F i 1 j 1 i 1 j 1 2 i 1 2 j 1 2 DOI: 10.9790/5728-12147286 www.iosrjournals.org 72 | Page The Exact Methods To Compute The Matrix Exponential 퐴푝푝푙푦푖푛푔 푡푕푖푠 푖푛푒푞푢푎푙푖푡푦 푡표 푎 푠푞푢푎푟푒 푚푎푡푟푖푥 퐴 푖푠 푒푎푠푦 푡표 푑푒푑푢푐푒 푡푕푒 푛푒푥푡 푓표푟푚: n A n A , for all n 1,2,3,.... F F Formally we must examine the convergence of the following limit: 푛 퐴푘 lim 푛 →∞ 푘! 푘=0 It is sufficient to observe that satisfies: k k n k n A n A A A F F e F k ! k ! k ! k 0 F k 0 k 0 and thus it demonstrated that 푒퐴 already is well defined for any square matrix with constant entry. It is useful to remember how the matrix exponential behaves under derivation: d t k A k (t ) dt k 0 k ! d tA t 2 A 2 t 3 A 3 t 4 A 4 t 5 A 5 I .......... ........ dt 1! 2! 3! 4! 5! 2 tA 2 3t 2 A 3 kt k 1 A k A .......... .. .......... ... 2! 3! k ! tA t 2 A 2 A I .......... 11 2! Ae tA e tA A. Using induction method and covenant condition (t ) (t ) follows the formula: k ( K ) d tA k tA tA k (2.1) ( t ) e A e e A ,k Z o dt k tA Note that the formula for the first derivative implies that the function x (t ) e x o is solution of initial value problem of the following first order system: x (t ) Ax , x (0 ) x o . Two results known of linear algebra that we used below are the following theorems. Theorem (2.1) Schur Triangularization Theorem. For any square matrix A n n , there is an unitary matrix U such that A UTU 1 is upper triangular. In addition, the entries in a diagonal matrix T are the eigenvalues of A . Theorem (2.2 ((Cayley Hamilton theorem) Let a square matrix and ( ) A I its characteristic polynomial then ( A ) 0 . Now we will discuss several methods by details to calculate or compute and find the matrix exponential. III. Diagonalizable Matrix Given a diagonal matrix n n D diag ( , ,......., ) It is to say that 1 2 n k k k k D diag ( 1 , 2 ,......... .......... n ) is true for all k Z then we have t k t k t k e tD D k diag k ,......., k 1 n k 0 k ! k 0 k ! k 0 k ! diag e 1 ,......, e n , DOI: 10.9790/5728-12147286 www.iosrjournals.org 73 | Page The Exact Methods To Compute The Matrix Exponential It is also a diagonal matrix. Now, in the case that A is a matrix diagonalizable it is known that there exists an invertible matrix P formed by the eigenvectors of A and a diagonal matrix D formed by the distinct eigenvalues of such that A PDP 1 . Now it is easy to verify the identity A k PD k P 1 for all k Z . Then we have: t k t k t k e tA A k ( PD k P 1 ) P D k P 1 k 0 k ! k 0 k ! k 0 k ! Pe tD P 1 . Accordingly, it is trivial to find the exponential matrix of a diagonalizable matrix, provided that previously we find all the eigenvalues of with corresponding eigenvectors. This case is well known. IV. Not Diagonalizable Matrix Suppose is not diagonalizable matrix which it is not possible to find n linearly independent eigenvectors of the matrix , In this case can use the Jordan form of . Suppose j is the Jordan form of , A j 1 with the transition matrix. Then e Pe P Where j diag ( j , j ,....., j ) diag ( j j ..... j ) 1 1 2 2 1 k 1 1 2 2 1 Then J j j j e (e 1 1 e 2 2 ...... e k k ) Thus, the problem is to find the matrix exponential of a Jordan block where the Jordan block has the form k J k ( ) k N k M k and in general N as ones on the k th upper diagonal and is the null matrix if k n the dimension of the matrix. By using the above expression we have k J ( ) 1 k 1 k 1 k k j j e k J ( I N ) N k 0 k ! k ! k 0 k ! j 0 J This can be written 2 n 1 j N N e e I N ....... 2! ( n 1)! Finally, we get the exponential matrix as follow: A j 1 e Pe P Then, tA t J t J 1 e Pdiag (e 1 ,........, e m ) P This case is also well known. One way to demonstrate this formula is to consider the system of first order x Jx with initial condition x (0 ) x o . On the one hand, we know that the solution of this system is given t J by x (t ) e x o . Furthermore, this system is easy to solve, starting last equation, which is decoupled, then every linear first-order equation is solved one by one, via the method of integrating factor. V. Triangular Matrix: Let S is an upper triangular matrix (lower triangular for a similar development is done) and write it as the sum of a diagonal matrix with a nilpotent matrix: S D N r Recall that a matrix N is called a nilpotent if there is a positive integer r such that N 0 . The smallest positive integer, for which this equality holds, is called the index of nilpotent of ownership matrix. Assuming the known property on exponential matrices see [1] e A B e A e B , if AB BA , Now, we can use the above formula e t s e t ( D N ) e t D e t N . to calculate: DOI: 10.9790/5728-12147286 www.iosrjournals.org 74 | Page The Exact Methods To Compute The Matrix Exponential We know how to calculate the matrix exponential of a diagonal matrix, so we discuss how to get e tN . Sufficient to note that when N nilpotent, the series of this matrix becomes finite, since the number of a quantity to be added to another is bounded by index of nilpotent of N .
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