I started looking at change of basis and had some question how it ties into eigenvalues/eigevectors.
If $T:R^n \to R^n, T(x) = Ax$, are eigenvalue/eigenvectors $A$ only defined (or only make sense) if the same basis is used to define the coordinates of vectors in both the domain and target space (i.e. the standard basis)? Otherwise I think it’s possible for a vector's coordinates to be scaled by the transformation but still change direction?
For the same matrix $A$, assuming it is diagonlizable ($A = X\Lambda X^{-1}$), is $X$ a change of basis matrix that takes a vector whose coordinates are defined in terms of the eigenbasis to the basis (not necessarily standard) for which $T$ is defined? If so, is the computation $y = X^{-1}AXx$ basically applying the transformation on $x$ to produce $y$, where the $X^{-1}$ and $X$ are added so both input and output vectors has coordinates in terms of the eigenbasis?