TITLE: Why is eigendecomposition $V \Lambda V^{-1}$ not $V^{-1} \Lambda V$
QUESTION [5 upvotes]: 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?

REPLY [4 votes]: The formula was generated from the equation $AV=V\Lambda$ which is a compact way of presentation a set of formulas $Av_i=\lambda_i{v_i}$ for eigenvectors.
In this case matrix $\Lambda$ is scaling column vectors ${v_i}$ grouped in the matrix $V=[v_1 \ \  v_2 \ \dots \ \ v_n]$.