Suppose we have a $n\times n$ real symmetric positive definite matrix $\Sigma$, and $V=(v_1,...,v_n)$ whose columns are the eigenvectors corresponding to the $n$ eigenvalues $\lambda_1\geq \lambda_2 ...\geq \lambda_n$. Let $\Delta$ be a diagonal matrix $diag(\delta_1,...,\delta_n)$, where $\delta_i>0$ for $i=1,...,n$. If $M=\Delta V$, then I want to ask that whether the columns of $M$ correspond to the eigenvectors of some matrix, saying $\tilde{\Sigma}$?
Many thanks!
$V$ is the eigenvector matrix, $D$ is the diagonal matrix.
$VD$ is still an eigenvector matrix. Multiplying a diagonal matrix from the right, scales the colums. Columns are eigenvectors. Scaled eigenvectors are still eigenvectors (of the same matrix that $V$ belongs to).
$DV$, is not. Multiplying a diagonal matrix from the left, scales the rows. This destroys the proportions of all the eigenvectors.
Regarding your last sentence, all vectors are the eigenvectors of the identity matrix, so that statement means very little.