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 DOI: 10.9790/5728-12147286 www.iosrjournals.org 72 | Page The Exact Methods to Compute The Matrix Exponential Mohammed Abdullah Salman 1, V. C. Borkar 2 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 tA e 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 observability 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 by tA e defined by . .......... )! 1 ( ) ( ...... ! 2 ) ( ! ) ( ) ( 1 0 2              n At At At I k At e t n k k At  To set the convergence of this series, we define firstly the frobenius norm of a matrix of size n m  as follow 2 1 2 1 1             m i n j ij F a A If ) (:, j A denotes the j-th column of A, and :) , ( i A the ith row, it is easy to see that is satisfy 2 1 1 2 2 2 1 2 2 1 :) . ( ) (:,                     m i n j F i A j A A 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 matrices n p p m B A   and then the product of them AB C ij  , with entries :) , ( :) , ( j B i A ij  . If A had complex entries, we obtain conjugate ij C applied to row :) , ( i A . Recall the Cauchy-Schwarz inequality 2 2 ) : , ( ) : , ( j B i A C ij  then we have : 2 2 F 2 2 1 2 2 1 2 2 2 2 1 1 2 1 1 2 j) B(:, ) : , ( ) j , (: ) : , ( F n j m i m i n j m i n j ij F B A i A B i A C AB               .