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Loser66
 one year ago
If \(A= \left[\begin{matrix}a&0\\b&c\end{matrix}\right]\)
1)compute \(e^{At}\)
2) Find the eigenvalues and eigenvectors of \(e^{A}\)
Please, help
Loser66
 one year ago
If \(A= \left[\begin{matrix}a&0\\b&c\end{matrix}\right]\) 1)compute \(e^{At}\) 2) Find the eigenvalues and eigenvectors of \(e^{A}\) Please, help

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Loser66
 one year ago
Best ResponseYou've already chosen the best response.0@oldrin.bataku I got \[e^{At}= \left[\begin{matrix}e^{at}(ac)&0\\b(e^{at}e^{ct})&e^{ct}(ac)\end{matrix}\right]\]

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0hence if t = 1, we have \(e^{A}= \left[\begin{matrix}e^{a}(ac)&0\\b(e^{a}e^{c})&e^{c}(ac)\end{matrix}\right]\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0for part 2), we know that \(e^{A}=(e^{A})^{1}\), so that we can calculate its eigenvalues and eigenvectors. However, it takes a long time to work on it. I would like to know there is any link between the two's eig.values and eig.vectors? Since we both work on the same A. Moreover, we have theorem for eigenvalue, it says if \(\lambda\) is one of eigenvalue of A, then \(\lambda^n\) is one of eigenvalue of \(A^n\) But I don't know whether we can apply for \(e^{A}\) and \(e^{A}\) or not, please, explain me

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0One more thing I got confuse: Surely \(A \neq A^{1}\), If I calculate \( A\) by putting  sign in the front of A, then, \(A = \left[\begin{matrix}a&0\\b&c\end{matrix}\right]\), while \[A^{1}= \left[\begin{matrix}1/c &0\\b/ac & 1/a\end{matrix}\right]\] How can \(e^{A}=(e^A)^{1}\) ???

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0by definition \(A,A\) commute so \(e^Ae^{A}=e^{A+A}=e^0=I\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0since \(e^{A}=(e^A)^{1}\) it follows the eigenvectors are identical but the eigenvalues are related by \(\lambda_{e^{A},i}=1/\lambda_{e^A,i}\)

Empty
 one year ago
Best ResponseYou've already chosen the best response.0Hmmm is there a reason why this method I'm using here doesn't work? First I separate A into a sum of the diagonal matrix D and the corner matrix C. \[A=D+C\] \[e^{At}=e^{Dt+Ct}=e^{Dt}e^{Ct}\] Then I compute them individually: \[e^{Dt} = \left[\begin{matrix}e^{at} & 0 \\ 0 & e^{ct} \end{matrix}\right]\] \[e^{Ct} = I+tC=\left[\begin{matrix}1 & 0 \\ bt & 1 \end{matrix}\right]\] Of course multiplying these together doesn't give the matrix I wanted since addition is commutative in the exponents but multiplying these two matrices together is easy to check that it's not commutative. I guess my most important question is, if \(C\) had been a nonsingular matrix, would this be valid, or is there more to it than that?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0well, you need \(C,D\) to commute to compute it in that way, and both being invertible is definitely insufficient  consider a change of basis followed by a scaling along the coordinate axes. these very clearly do not commute as linear transformations and yet both are invertible

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0the standard way to compute \(\exp(A)\) for diagonalizable \(A\) is to diagonalize \(A=P^{1}DP\) where \(P\) rotates into an eigenbasis and \(D\) describes the scaling, since \(A^n=(P^{1}DP)^n=P^{1}D^nP\) so: $$\exp(A)=P^{1}\left(\sum_{n=0}^\infty\frac1{n!}D^n\right)P$$ and \(D^n\) is trivial for diagonal matrices

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@empty if you're curious as to when two matrices A,B commute: https://en.wikipedia.org/wiki/Commuting_matrices#Characterization_in_terms_of_eigenvectors

Empty
 one year ago
Best ResponseYou've already chosen the best response.0Yeah I'm only able to know when matrices commute when I understand their geometric interpretation such as rotation matrices will commute with each other and with scalar matrices, things like that. Thanks @oldrin.bataku
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