### Transcription

International Journal of Scientific and Innovative Mathematical Research (IJSIMR)Volume 4, Issue 1, January 2016, PP 53-63ISSN 2347-307X (Print) & ISSN 2347-3142 (Online)www.arcjournals.orgExponential Matrix and Their PropertiesMohammed Abdullah Saleh Salman1,2Dr. V.C.BorkarCollege of Education & languages,Department of Mathematics & Statistics,University of Amran.Amran, [email protected] Mahavidyalaya,Department of Mathematics & Statistics,Swami Ramanand Teerth MarthwadaUniversity, Nanded, [email protected]: The matrix exponential is a very important subclass of matrix functions. In this paper, we discusssome of the more common matrix exponential and some methods for computing it. In principle, the matrixexponential could be calculated in different methods some of the methods are preferable to others butnone are entirely satisfactory. Due to that, we discussed computations of the matrix exponential using TaylorSeries, Scaling and Squaring, Eigenvectors, and the Schur decomposition methods theoretically.Keywords: Matrix Exponential, Commuting Matrix, Non-commuting Matrix.1. INTRODUCTIONThe purpose of this note is matrix functions, The theory of matrix functions was subsequentlydeveloped by many mathematicians over the ensuing 100 years. Today, matrices of functions arewidely used in science and engineering and are of growing interest, due to the succinct way theyallow 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 areasespecially matrix Exponential .The matrix exponential is a very important subclass of functions ofmatrices that has been studied extensively in the last 50 years [ ]. The computation of matrixfunctions has been one of the most challenging problems in numerical linear algebra. Among thematrix functions one of the most interesting is the matrix exponential. A large number of methods hasbeen proposed for the matrix exponential, many of them of pedagogic interest only or of dubiousnumerical stability. Some of the more computationally useful methods are surveyed in [ ] Inprinciple, the matrix exponential could be computed in many ways and many different methods tocalculate matrix exponential [ ,9]. In practice, some of the methods are preferable to others, but noneare completely satisfactory.2. DEFINITIONS OF EXP(A):The functions of a matrix in which we are interested can be defined in various ways. Inmathematics, the matrix exponential is a function on square matrices analogous to the ordinaryexponential function [1, , , , 7]. Let A Mn. The exponential of A, denoted by e A orexp(A) , is the n n matrix given by the power serieseAk 0Akk!IAA22!.An 1(n 1)!( 1)Where A0 INote that this is the generalization of the Taylor series expansion of the standard Exponentialn nfunction. The series (1) converges absolutely for all A Chas radius of convergenceequal to 1), so the exponential of A is well-defined. To prove the Convergence of the series,we have the following theorem. ARCPage 53

Mohammed Abdullah Saleh Salman & Dr. V.C.BorkarTheorem (2.1) for more detail in [6]:The series (1) converges absolutely for all AM n . Furthermore, letbe a normalized submultiplicative norm on M n . TheneAeA(2)Proof:The n th partial sum isAkk!Snk 0SoeASnk 0Since AnumbersAkk!Akk!mk 0k m 1AkAkk!Ak!k m 1k m 1kk!is a real number and the right-hand side is a part of the convergent series of realekthen this equation is convergent, ifekAA0k! 0 there is an N such that for m n,AAkmkk!1This is sufficient to prove that S n is convergent. Furthermore, note thateAk0AkAkk!k0k!k0Ak!keAn n complexIn some cases, it is a simple matter to express the matrix exponential of anmatrix A shall be denoted by ee At12 iAand can be defined in a number of equivalent ways [ ]:e zt ( zIA) 1 dz(3)Ore Atlim k(1At k)k(4)Ore AtdxdtAX (t ) ,X(0)1(5)For details see [7], and we have other definitions but we leave it to reader to collect them.3. COMPUTATION OF EXPONENTIAL MATRIXThere are many methods used to compute the exponential of a matrix. Approximation Theory,differential equations, the matrix eigenvalues, and the matrix characteristic Polynomials aresome of the various methods used. we will outline various simplistic Methods for finding theexponential 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 Properties3.1- Computing Matrix Exponential for Diagonal Matrix and for Diagonalizable Matricesif A is a diagonal matrix having diagonal entries then we havee a1ea 2eA ea nEigenvectorsRnANow, Let benv1 , v 2 ,., v nAv ksymmetric and has a complete set of linear independentsuch thatk vkk 1,2,. .nlet us define the matrix T [ v1 , v 2 ,.,(6)v n ] whose columns are the eigenvectorof A corresponding to the eigenvalues of A, we haveATAv1, Av2 ,., Avnv1, v2 ,. v3T1Since AT T1where A2 nNow using AT T1to compute e A and we can write it as followeAk 0Akk!1(TΛT 1 ) kk!T(k 01 kΛ )Tk!1TeΛ T1And henceeeA1eT2T e1nExample:Consider the matrixA3 0 00 5 00 0 1then by using the above formula for diagonal form we get the exponential matrix ise3 0 0eA0 e5 00 0 e1For diagonalizable matrix we give this exampleInternational Journal of Scientific and Innovative Mathematical Research (IJSIMR)Page 55

Mohammed Abdullah Saleh Salman & Dr. V.C.BorkarExample:Let5A12 2after found the eigenvalues and eigenvectors and construct matrix T we use this formulaA T T 1 to compute e A as follow1eAe4 011 -22e301-112e 4 - e 3e4e32e 32e 3e42e 43.2- Computing Matrix Exponential for General Square Matrices3.2.1- Using Jordan Normal FormSuppose A is not diagonalizable matrix which it is not possible to find n linearly independenteigenvectors of the matrix A, In this case can use the Jordan form of A. Suppose j is theJordan form of A, with P the transition matrix. TheneATe jT1Wherejdiag ( j1 1 , j2 2 ,., j1 k ) diag ( j11j22.j1 )Then(e j1 1eJe j2e jk k ).2Thus, the problem is to find the matrix exponential of a Jordan block where the Jordan blockN k M k and in general N k as ones on the k thhas the form J k ( )upperkdiagonal and is the null matrix if k n the dimension of the matrix. by using the aboveexpression we haveeJ k ()k01 kJk!k01( Ik!N )kk01 k kk! j 0 JkjNjThis can be writtenejeINN22!.Nn 1(n 1)!Example:A21 1765 -1 - 644 16Then we calculate the eigenvalues of A which are [ ] We have Awhich isP1241-2404PJP1then we calculate P54140AndInternational Journal of Scientific and Innovative Mathematical Research (IJSIMR)Page 56

Exponential Matrix and Their PropertiesJ16 10 16(4)Therefore, by using the Jordan canonical form to compute the exponential of matrix A is13e16eA14e49e16e413e 165e 4- 9e165e 416e162e16- 2e1616e 162e 42e 44e163.2.2- Using Hamilton Theorem CayleyTheorem 3.1 (Cayley Hamilton)Let A a square matrix and(A)( )AIits characteristic polynomial then0.Proof:Consider a n n square matrix A and a polynomial p(x) andpolynomial of A. Then write p(x) in the formp ( x)by Cayley-Hamilton(x)(x) be the characteristic( x) q( x) r ( x)0 , then p(A) r(A) such that we can write polynomial1 kXk!rk ( X )Where rk (x ) is the remainder of long division ofxkbyk!(x) , Then the matrix exponentialcan be written asn 1eAk 0Akk!rk ( A)k 0thus e A is a polynomial of A of degree less than nkn 1eAak Ak 0Consider now an eigenvector v with the corresponding eigenvaluee Avk1 kA vk!0k, Then1 kv e vk!0Analogouslyn 1n 1kak A v(k 0akk)vk 0and thus if we have n distinct eigenvaluesso that the above equation is an interpolationproblem which can be used to compute the coefficients a k . In the case of multiple eigenvalueswe use the corresponding generalized eigenvectors.3.2.3- Using Numerical IntegrationConsider the ODE xk Ax k ,solution from 1 to n we getx(0)ek(0,0,.0, 1,.0,0) T then when collect theInternational Journal of Scientific and Innovative Mathematical Research (IJSIMR)Page 57

Mohammed Abdullah Saleh Salman & Dr. V.C.Borkar(e tA e1 ,., e tA e2 )x1 (t ),. ., x n (t )e tA (e1 ,., en )e tA Ie tAthen the general solution for above ODE ise tAX (t )XkXk1ttwith X 0mtNow, by using numerical integrator with step(t k , X k ), kI we get0,., m - 1That impliesetAXm3.2.4-The Matrix Exponential Via InterpolationHere we have two kinds as follow:3.2.4. A-Lagrange Interpolation FormulaLet1,2,.,nbe the distinct eigenvalues of a matrix AM n and f(t) is any function that iswell defined at the eigenvalues of A, then the Lagrange formula for e A iskke tAeitij 1,i jAiIij.(7)3.2.4.B- Newton's Divided Difference InterpolationLet AM n be a matrix with eigenvalues( A)1,2 ,.,nNow we define f(A) as followsi 1nf ( A)f [ 1,2 ,.,]i 1Where [ 1 ,2,.,nf [ 1 ,. . .,(A] is the divided difference ati]iI)(8)j 1f [ 1 ,. . .,i]1f[n,2,.,nwhich defined2 ,. . .,n 1]n 11where the value of divided difference is independent of the order of the arguments.3,2.5- Using a Limit of PowerFrom calculus we know that for any numbers a and t the exponentiale atlim k(1A k)k(9)from equation (4)one can define the matrix exponential as a limit of powers aseAlim k(1A k)kThis formula is the limit of the first order Taylor expansion of(10)Araised to the power nnZ.4. SCALING AND SQUARINGWe derive a scaling property from a fundamental nonlinear differential equation whosesolution is the so-called q-exponential function. A scaling property has been believed to begiven by a power function only, but actually more general expression for the scaling propertyis found to be a solution of the above fundamental non-linear differential equation. ThisInternational Journal of Scientific and Innovative Mathematical Research (IJSIMR)Page 58

Exponential Matrix and Their Propertiesmethod will help to control some of the round off error and time or number of terms it wouldtake to find a Taylor approximation The scaling and squaring method is the most widely usedmethod for computing the matrix exponential. The method scales the matrix by a power of 2 toreduce the norm to order 1.The advantage of the scaling methods is that the scaled transitionmatrix can be made to have a norm less than unity.5. TAYLOR SERIESLet A M n The exponential of A, denoted by e A or exp (A), is thethe Taylor power serieseAk 0Where A0Akk!IAA22!An 1(n 1)!.n n matrix given by(11)I.Note that this is the generalization of the Taylor series expansion of the standard exponentialfunction. The above series always converges and well-defined. Now to calculate the matrixexponential we use computers and we cannot calculate the exponential matrix exact only wewill be able to approximate it with a truncated Taylor series of k terms. The truncated Taylorseries is denoted by Rk ( A) the order of the approximation is seen to be the highest power ofthe truncated Taylor series which represented by k but there are many other factors that canaffect the accuracy of a solution technique such that For accuracy and time efficiency, theresult for the matrix exponential depends on the matrix norm when the matrix norm is verylarge then the turn may cause in accuracy due to numerical round off this is problem occurswhen the entries of A are large ,otherwise if the norm is small then accuracy and timeefficiency as we desired.6. EIGENVECTORS AND SCHUR DECOMPOSITION METHODSThis method based on the similarity transformation of a matrix as followeAPe P1where A is a real symmetric matrix and P a real unitary matrix andthe eigenvalues of Awhich are real, this is easy to compute when A is non defective (diagonalizable) but, when Amay not be diagonalizable and thus is defective such that there is no invertible matrix ofeigenvectors P. When P is not invertible then it has ill conditioned so the error will be large.Due to these observations this method relies on diagonalizating the matrix.7. PROPERTIESIn this section of this paper we collect for reference additional important properties of thematrix exponential that are not needed in the development [4,8,9]. Let A, B M n and let t ands be arbitrary complex numbers. We denote the n n Zero matrix by 0. The matrixexponential satisfies the following properties0Property (1) If 0 denotes the zero matrix, then eProperty (2) If A is invertible, then e ABAProperty (3) if A1Ae B A -1.diag( A1 , A2, ., Ak ), then e AAProperty (4) det(e )I the identity matrix.(eA1 ,.,e Ak ) .e trac ( A) . when A is complex square matrix and trace(A) 0 thendet(e A ) 1.( AT )(e A )T . it follows that if A is symmetric, then e A is alsoAsymmetric, and if A is skew symmetric then e is orthogonal.Property (5) eInternational Journal of Scientific and Innovative Mathematical Research (IJSIMR)Page 59

Mohammed Abdullah Saleh Salman & Dr. V.C.BorkarBe B A and e A e B e B e A unfortunately not allProperty (6) if AB BA then Aefamiliar properties of the scalar exponential function y e t carry over to the matrixexponential. For example, we know from calculus e s t e s e t when s and t arenumbers. However this is often not true for exponentials of matrices. In other words, itAis possible to have n n matrices A and B such that eA BAe ehave equality ediscussed in this section.PropertymA eA(7)Ae ABe A e B . Exactly when wedepends on specific properties of the matrix A and B thatletmBAbeacomplexn nsquareMatrixthenfor integer m.Property (8) Let A be a complex square matrix, and let s, tC Then e A( st)e As e At .0I . In other words, regardless of theSetting s 1 and t -1 in property (1) we get eAmatrix A, the matrix exponential e is always invertible, and has inverse e A .(e A ) . it follows that if A is Hermitian matrix, then e A is alsoHermitian, and if A is skew-hermitian, then e A is.Property (9)e( A )Property (10) (e At )Property (11)e( AB )tAe tA .let A, BM n then ABBA if an