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I have a 3x3 transformation matrix $D_{BB} (f)$ with $B$ as a basis of vector space $V$ and $f$ as a diagonalizeable endomorphism $f : V \to V$ given. Basis $B$ is not explicitly given. The entries of $D_{BB} (f)$ are simply integers. A function for $f$ is also not explicitly given, though it is diagonalizeable (as $D_{BB} (f)$ has n distinct eigenvectors). I have another basis $B'$ of $V$, which is composed of the eigenvectors. These I have calculated, so I do have an explicit $B'$.

For the assignment I'm supposed to calculate the transformation matrices $D_{BB'}(Id_V)$ $D_{B'B}(Id_V)$. I have no idea how to do this without knowing anything explicit about $f$ or $B$. Is there a way to calculate B from $D_{BB} (f)$? Do I need to know $B$ or $f$ or is there a way around the lack of explicitly declared vectors and/or a function?

I saw a formula with $D_{B'B'}(f) = D_{B'B}(f)*D_{BB}(f)*(D_{B'B}(f))^{-1}$ but without an explicit $f$ to calculate $D_{B'B'}(f)$ I'm not sure how I would use that either.

Thanks for any help/ideas/nudges in the right direction.

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So if I understand correctly, you have calculated the eigenvectors from the matrix $D_{BB}(f)$? Then $D_{BB'}(Id_V)$ is in fact the matrix with columns the eigenvectors you have calculated, and it's inverse is $D_{B'B}(Id_V)$.

The formula you have should be $D_{B'B'}(f)=D_{B'B}(Id_V)D_{BB}(f)D_{B'B}(Id_V)$. So the matrices $D_{BB'}(Id_V)$ and $D_{B'B}(Id_V)$ are actually change-of-basis matrices, the former from $B'$ to $B$ and the latter the other way round.

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