Say you have a linear transformation matrix $A$. In the basis of eigenvectors, this transformation simply becomes a scaling, represented by the diagonal matrix of eigenvalues.
Thus, intuitively the transformation A can be decomposed into the following:
- Transform into the basis of eigenvectors (using the transformation matrix $V$, where the eigenvectors form the columns)
- Apply the scaling.
- Transform back.
This would seem to correspond to $V^{-1} \Lambda V$, where the standard notation of matrices being applied on the left and vectors on the right holds.
Yet everywhere I always see the formula as $V \Lambda V^{-1}$. Why isn't my intuition correct?