2MA 36600 LECTURE NOTES: WEDNESDAY, APRIL 22Now we return to the general case. Consider ann×nmatrixA. Say that there exists a nonsingularn×nmatrixTsuch thatT−1A T=Dis ann×ndiagonal matrix. Such a matrixAis said to bediagonalizable.We will show thatexp (At) =Texp (Dt)T−1.In practice this is easy to compute because bothT−1and exp (Dt) are easy to compute.Observe thatT−1AkT=Dkfor any nonnegative integerk:k= 0:T−1A0T=T−1I T=I=D0k= 1:T−1A1T=D1k≥2:T−1AkT=T−1Ak−1T T−1A T=T−1Ak−1TT−1A T=Dk−1D=DkThis gives the expressionT−1exp (At)T=∞k=0T−1AkTtkk!=∞k=0Dktkk!= exp (Dt).The claim follows.Example.Let us review the following homogeneous system considered in the previous lecture:ddtx=1141x.We found that the general solution isx(t) =c112e3t+c21−2e−t=Ψ(t)cin terms ofΨ(t) =e3te−t2e3t−2e−t,c=c1c2.We explain how to find the fundamental matrixΨ(t) using exponentials of matrices.Denote the 2×2 matricesA=1141andΨ0=112−2.The fundamental matrix above is the unique solution to the initial value problemddtΨ=AΨ,Ψ(0) =Ψ0.We know that this has the unique solution
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