# ch8 - CHAPTER 8 MATRIX EXPONENTIAL METHODS SECTION 8.1 In...

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416Chapter 8CHAPTER 8MATRIX EXPONENTIAL METHODSSECTION 8.1In Problems 1–8 we first use the eigenvalues and eigenvectors of the coefficient matrixAtofind first afundamental matrixΦ(t)for the homogeneous systemx=Ax.Then we apply theformulax(t)=Φ(t)Φ(0)-1x0,to find the solution vectorx(t)that satisfies the initial conditionx(0)=x0.Formulas (11) and(12) in the text provide inverses of2-by-2and3-by-3matrices.1.Eigensystem:TT11221,[11] ;3,[11]λλ====vv123123( )tttttteeteeeeλλΦ==vv3333113511( )112225tttttttteeeeteeee+ == + x2.Eigensystem:TT11220,[12] ;4,[12]λλ====vv1241241( )22tttteteeeλλΦ==vv444421213511( )2114422610tttteetee+ ==  x3.Eigensystem:T4 ,[122]iiλ==+vcos42sin42cos4sin4( )Re()Im()2cos42sin 4ttttttteettλλ+Φ==vvcos42sin42cos4sin40205sin 411( )2cos42sin42114cos42sin444tttttttttt+  ==    x
Section 8.14174.Eigensystem:TT12122, 2;{,} with[11] ,[10]λ===vvvv211211( )()1ttttteteetλλ+Φ=+=vvv22110111( )1110ttttteett++   ==      x5.Eigensystem:T3 ,[ 13]iiλ== − +vcos3sin3cos3sin3( )Re()Im()3cos33sin3ttttttteettλλΦ==vvcos3sin3cos3sin30113cos3sin311( )3cos33sin33113cos36sin333ttttttttttt == + x6.Eigensystem:T54 ,[122]iiλ=+=+v5cos42sin42cos42sin4( )Re()Im()2cos42sin 4tttttttteeettλλ+Φ==vv55cos42sin42cos42sin4022cos4sin 41( )22cos42sin 4240sin44ttttttttteettt++  ==    x7.Eigensystem:TTT1122330,[625] ;1,[312] ;1,[212]λλλ===== −=vvv312123632( )2522ttttttttteeteeeeeeeλλλΦ==vvv632021212122( )212214452213001082tttttttttttteeeeteeeeeeee++    ==++  

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