What is my change of basis matrix?

Question

When we are diagonal a matrix we do the following: $$P^{-1}MP$$ Where $P$ is the matrix with columns as the eigenvectors of $M$. Let us say that $M$ is representing some linear map in the basis $E$ and we want to convert it to the egienbais F. If we have a vector $[v]_E$ in the bias E is $P^{-1}$ the change of bais matrix i.e.: $$[v]_F=P^{-1}[v]_E$$ or is it $P$ i.e. $$[v]_F=P[v]_E$$ I think it is the former but am not 100% sure.

Answer 1

If $E$ is the given basis and $F$ is the eigenbasis and $[v]_E$ is the coordinate vector of $v$ relative to $E$ and $[v]_F$ is the coordinate vector of $v$ relative to $F$, then we have $[v]_E = P[v]_F$ or equivalently $[v]_F = P^{-1}[v]_E$.

This is not too difficult to remember. Just remind yourself that, in the _eigen_basis $F$, the coordinate vectors of the eigenvectors are unit vectors, e.g. $(0,\ldots,0,1,0,\ldots,0)$, that's true simply because the eigenvectors are basis elements. Now, as you said yourself, the columns of $P$ are the coordinate vectors of the eigenvectors of $M$ in the given basis $E$. So if we multiply $P$ from the right with a unit vector, we get the coordinate vector of an eigenvector of $M$ in the given basis $E$. So the transformation rule must be $[v]_E= P[v]_F.$