Question

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

Answer 1

@oldrin.bataku
I got
\[e^{At}= \left[\begin{matrix}e^{at}(a-c)&0\\b(e^{at}-e^{ct})&e^{ct}(a-c)\end{matrix}\right]\]

Answer 2

hence if t = 1, we have \(e^{A}= \left[\begin{matrix}e^{a}(a-c)&0\\b(e^{a}-e^{c})&e^{c}(a-c)\end{matrix}\right]\)

Answer 3

for 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

Answer 4

One 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 \(e^{-A}=(e^A)^{-1}\) ???

Answer 5

by definition \(-A,A\) commute so \(e^Ae^{-A}=e^{A+-A}=e^0=I\)

Answer 6

since \(e^{-A}=(e^A)^{-1}\) it follows the eigenvectors are identical but the eigenvalues are related by \(\lambda_{e^{-A},i}=1/\lambda_{e^A,i}\)

Answer 7

Hmmm 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?

Answer 8

well, 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

Answer 9

the 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

Answer 10

@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

Answer 11

Yeah 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