I'm trying to get a eigenmatrix of the form:
$ \Lambda = \begin{bmatrix} \frac{1}{\sqrt{\lambda_1}} & 0 & \dots & 0 \\ 0 & \frac{1}{\sqrt{\lambda_2}} & \dots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & \dots & 0 & \frac{1}{\sqrt{\lambda_n}} \end{bmatrix} $
Though, I am having issues deriving it. Best I can think of is to have
$ Ax = \lambda_i x \implies \frac{1}{\lambda_i}Ax = x. $
Then, from this, to perform a decomposition of the left hand side to get the desired eigenmatrix $\Lambda$.
Any help is appreciated.