International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 4, Issue 1, January 2016, PP 53-63 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org Exponential Matrix and Their Properties Mohammed Abdullah Saleh Salman1,2 Dr. V.C.Borkar College of Education & languages, Yeshwant Mahavidyalaya, Department of Mathematics & Statistics, Department of Mathematics & Statistics, University of Amran. Swami Ramanand Teerth Marthwada Amran, Yemen. University, Nanded, India [email protected] [email protected] Abstract: The matrix exponential is a very important subclass of matrix functions. In this paper, we discuss some of the more common matrix exponential and some methods for computing it. In principle, the matrix exponential could be calculated in different methods some of the methods are preferable to others but none are entirely satisfactory. Due to that, we discussed computations of the matrix exponential using Taylor Series, Scaling and Squaring, Eigenvectors, and the Schur decomposition methods theoretically. Keywords: Matrix Exponential, Commuting Matrix, Non-commuting Matrix. 1. INTRODUCTION The purpose of this note is matrix functions, The theory of matrix functions was subsequently developed by many mathematicians over the ensuing 100 years. Today, matrices of functions are widely used in science and engineering and are of growing interest, due to the succinct way they allow solutions to be expressed and recent advances in numerical algorithms for computing them [ ]. In general is an interesting area in linear algebra, matrix analysis and are used in many areas especially matrix Exponential .The matrix exponential is a very important subclass of functions of matrices that has been studied extensively in the last 50 years [ ]. The computation of matrix functions has been one of the most challenging problems in numerical linear algebra. Among the matrix functions one of the most interesting is the matrix exponential. A large number of methods has been proposed for the matrix exponential, many of them of pedagogic interest only or of dubious numerical stability. Some of the more computationally useful methods are surveyed in [ ] In principle, the matrix exponential could be computed in many ways and many different methods to calculate matrix exponential [ ,9]. In practice, some of the methods are preferable to others, but none are completely satisfactory. 2. DEFINITIONS OF EXP(A): The functions of a matrix in which we are interested can be defined in various ways. In mathematics, the matrix exponential is a function on square matrices analogous to the ordinary A exponential function [1, , , , 7]. Let A ∈ Mn. The exponential of A, denoted by e or exp(A) , is the n × n matrix given by the power series k 2 n 1 A A A A e I A ...... (`1) k 0 k! 2! (n 1)! Where A0 = I Note that this is the generalization of the Taylor series expansion of the standard Exponential n n function. The series (1) converges absolutely for all A C has radius of convergence equal to +1), so the exponential of A is well-defined. To prove the Convergence of the series, we have the following theorem. ©ARC Page | 53 Mohammed Abdullah Saleh Salman & Dr. V.C.Borkar Theorem (2.1) for more detail in [6]: The series (1) converges absolutely for all A M n . Furthermore, let be a normalized sub multiplicative norm on M n . Then eA e A (2) Proof: The nth partial sum is Ak Sn k 0 k! So k k Ak m Ak Ak A A e A S n k! k! k! k! k! k 0 k 0 k m 1 k m 1 k m 1 Since A is a real number and the right-hand side is a part of the convergent series of real numbers A k e A k 0 k! then this equation is convergent, if > 0 there is an N such that for m > n, A k e A k! k m 1 This is sufficient to prove that S n is convergent. Furthermore, note that k Ak k A A A A e e k 0 k! k 0 k! k 0 k! In some cases, it is a simple matter to express the matrix exponential of an n n complex A matrix A shall be denoted by e and can be defined in a number of equivalent ways [ ]: 1 e At e zt (zI A) 1 dz (3) 2 i Or At At k e lim (1 ) (4) k k Or At dx e AX(t) , X(0) 1 (5) dt For details see [7], and we have other definitions but we leave it to reader to collect them. 3. COMPUTATION OF EXPONENTIAL MATRIX There are many methods used to compute the exponential of a matrix. Approximation Theory, differential equations, the matrix eigenvalues, and the matrix characteristic Polynomials are some of the various methods used. we will outline various simplistic Methods for finding the exponential of a matrix. The methods examined are given by the type of matrix [ , ,8,9]. International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 54 Exponential Matrix and Their Properties 3.1- Computing Matrix Exponential for Diagonal Matrix and for Diagonalizable Matrices if A is a diagonal matrix having diagonal entries then we have ea1 ea 2 e A a e n Now, Let be A R n n symmetric and has a complete set of linear independent Eigenvectors v1 ,v2 ,......., vn such that Av k k vk k 1,2,...........n (6) let us define the matrix T = [ ] whose columns are the eigenvector of A corresponding to the eigenvalues of A, we have AT Av1, Av 2 ,......., Avn v 1 , v 2 ,........ v 3 T 1 Since A T T 1 where A 2 n Now using to compute and we can write it as follow Ak 1 1 eA (TΛT 1)k T( Λk )T 1 TeΛT 1 k 0 k! k! k 0 k! And hence e 1 e 2 e A T T 1 e A e n Example: Consider the matrix 3 0 0 A 0 5 0 0 0 1 then by using the above formula for diagonal form we get the exponential matrix is e3 0 0 e A 0 e5 0 0 0 e1 For diagonalizable matrix we give this example International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 55 Mohammed Abdullah Saleh Salman & Dr. V.C.Borkar Example: Let 5 1 A 2 2 after found the eigenvalues and eigenvectors and construct matrix T we use this formula to compute as follow 1 1 e4 0 2 1 2e 4 - e3 e4 e3 e A 1 - 2 0 e3 1 -1 2e3 2e4 2e 3 e4 3.2- Computing Matrix Exponential for General Square Matrices 3.2.1- Using Jordan Normal Form Suppose A is not diagonalizable matrix which it is not possible to find n linearly independent eigenvectors of the matrix A, In this case can use the Jordan form of A. Suppose j is the Jordan form of A, with P the transition matrix. Then A j 1 e Te T Where j diag( j1 1, j2 2 ,....., j1 k ) diag( j1 1 j2 2 ..... j1 ) 1 AThenT T 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 k has the form Jk ( ) k Nk 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 k J k ( ) 1 k 1 k 1 k j j e J ( I N) N k 0 k! k 0 k! k 0 k! j 0 J This can be written 2 n 1 j A N N e e e I N ....... 2! (n 1)! Example: 21 17 6 A 5 -1 - 6 4 4 16 Then we calculate the eigenvalues of A which are [ ] We have A PJP 1 then we calculate P which is 1 5 2 4 4 1 1 P - 2 4 4 0 4 0 And International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 56 Exponential Matrix and Their Properties 16 1 J (4) 0 16 Therefore, by using the Jordan canonical form to compute the exponential of matrix A is 13e16 e4 13e16 5e4 2e16 2e4 1 e A 9e16 e4 - 9e16 5e4 - 2e16 2e4 4 16e16 16e16 4e16 3.2.2- Using Hamilton Theorem Cayley Theorem 3.1 (Cayley Hamilton) Let A a square matrix and ( ) A I its characteristic polynomial then (A) 0. Proof: Consider a n n square matrix A and a polynomial p(x) and (x) be the characteristic polynomial of A. Then write p(x) in the form p(x) (x)q(x) r(x) by Cayley-Hamilton (x) 0 , then p(A) = r(A) such that we can write polynomial 1 X k r (X ) k! k x k Where r (x) is the remainder of long division of by , Then the matrix exponential k k! can be written as n 1 Ak e A r (A) k! k k 0 k 0 thus e A is a polynomial of A of degree less than n n 1 k A e ak A k 0 Consider now an eigenvector v with the corresponding eigenvalue , Then A 1 k 1 k e v A v v e v k 0 k! k 0 k! Analogously n 1 n 1 k k ak A v ( ak )v k 0 k 0 and thus if we have n distinct eigenvalues so that the above equation is an interpolation problem which can be used to compute the coefficients a k .