Chapter 8Systems of Linear First-Order DEs 8.4 Matrix Exponential MAT 246 Differential Equations Review Material Appendix II.1 (Definition II.10 and II.11) 8.4 Matrix Exponential Erickson/Biddy2 Homogeneous Systems It is possible to define a matrix exponential

so that (1) is a solution of the homogeneous solution X = AX. A is an n n matrix of constants C is an n 1 column matrix of arbitrary constants. Erickson/Biddy 8.4 Matrix Exponential 3 te =AX CLinear Algebra Review Special Matrices Identity Matrix 5.3 Matricies and Linear Systems 4 1 0 00 1 00 0 1 ( ( (= ( ( nILinear Algebra Review Inverse The inverse of a square matrix A is a square matrix B s.t. AB = BA = I. If a matrix has an inverse it is unique and denoted by A-1. i.e.5.3 Matricies and Linear Systems 5 -1 -1AA = A A= ILinear Algebra Review Computing Inverses Computing inverses for a 2x2 matrix. For larger matrices we do row operations.AI = IA-1 5.3 Matricies and Linear Systems 6 1| det |a bc dd bc a (=( (= ( -1AAALinear Algebra Review Inverses A-1 exists iff det(A)=0 (A is non-singular matrix) A-1 does not exist if det(A)=0 (A is a singular matrix) 5.3 Matricies and Linear Systems 7 Linear Algebra Review Diagonal Matrix An n x n matrix A is a diagonal matrix if all its entries off the main diagonal are zero, that is,Erickson/Biddy 8.4 Matrix Exponential 8 11220 00 00 0nnaaa ( ( (= ( ( ALinear Algebra Review Integral of a Matrix of Functions If A(t) = (aij(t))mn is a matrix whose entries are functions continuous on a common interval containing t an t0, then Erickson/Biddy 8.4 Matrix Exponential 9 n mttijttds s a ds s||.|

\|= } }0 0) ( ) ( ADefinition 8.4.1 Matrix Exponential For any n n matrix, A, This is equation (3). It can be shown that the series given in (3) converges to an n n matrix for every value of t. Erickson/Biddy 8.4 Matrix Exponential 10 Example pg. 359 Use (3) to compute eAt and e-At.Erickson/Biddy 8.4 Matrix Exponential 11 0 11 0 (=( 2.ADerivative of eAt The derivative of the matrix exponential is (4)Erickson/Biddy 8.4 Matrix Exponential 12 t tde edt=A AAFundamental Matrix Given the system X = AX its general solution is X = c1X1 + c2X2 + + cnXn Which can be written as X = (t)C, where C is a n1 column matrix of arbitrary constants and (t) is the nn matrix whose columns consist of the entries of the solution vectors of the system Erickson/Biddy 8.4 Matrix Exponential 13 (((((

=nn n nnnx x xx x xx x xt 2 12 22 211 12 11) ( eAt is a Fundamental Matrix If we denote the matrix exponential eAt by the symbol (t), then (4) is equivalent to the matrix DE(t) = A(t). It follows from Definition 8.4.1 that (0) = eA(0) = I, and so det (0) 0. These two properties are enough for us to conclude that (t) is a fundamental matrix of the system X = AX. Erickson/Biddy 8.4 Matrix Exponential 14 Example pg. 359 Use (1) to find the general solution of the given system. Erickson/Biddy 8.4 Matrix Exponential 15 0 11 0 (' =( 6.X XNonhomogeneous Systems For a nonhomogeneous system of linear first-order DEs it can be shown that the general solution of X = AX + F(t), where A is an n n matrix of constants, is (5) Since the matrix exponential eAt is a fundamental matrix, it is always nonsingular and e-At = (eAt)-1. In practice, e-As can be obtained from eAt by simply replacing t by s. Erickson/Biddy 8.4 Matrix Exponential 16 Examples pg. 359 Use (5) to find the general solution of the given system. Erickson/Biddy 8.4 Matrix Exponential 17 41 00 20 1 cosh1 0 sinhttett ((' = + (( ((' = + (( 10.X X12.X XUse of the Laplace Transform We also have Erickson/Biddy 8.4 Matrix Exponential 18 ( ){ }11 te s= AI A LExample pg. 359 Use the method of Example 2 (using Laplace transforms) to compute eAt for the coefficient matrix.Use (1) to find the general solution of the given system.Erickson/Biddy 8.4 Matrix Exponential 19 4 21 1 (' =( 16.X X