Chapter 4 The Fundamental Matrix Consider the linear differential equation system X(t)=AX(t) where X is the initial state vector,X(t)eR",Ae consth(n,n) Multiply both sides by e-Ar eAx()=eAAX(t) According to Proposition 4 eA=-e(←A)=-de dt ew4x(+4ex()=0 dt dt According to the product rule for differentiation of matrix function (ex())=0 dt SO eX(①=C, CERT Vector C is decided by the initial condition: C=e-A X(to) so etx①=e点Xt) Considering e-A=(e4),we have X(t.X)=eNle-Ax(t) This is the solution of system.Chapter 4 8