Suppose M is normal and has distinct eigenvalues $\lambda_1,\dots,\lambda_n$. Then this means the rational canonical form of $M$ is one single full block companion matrix
$\begin{pmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ a_0 & a_1 & a_2 & \dots & a_{n-1} \end{pmatrix}$
and the eigenvectors of this companion matrix is the Vandermonde matrix $V$ in $\lambda_1,\dots,\lambda_n$.
Is it true that I can write $M=U\text{diag}(\lambda_1,\dots,\lambda_n)U^{-1}$ where $U$ is specifically the unitary part of the unique polar decomposition of the Vandermonde $V=HU$, with $H:=\sqrt{VV^\ast}$ and $U:=H^{-1}V$?? I don't know how to show this.
I know there are varying conventions to some definitions I used, since mine are reasonably standard I hope comments here can keep to the posted conventions.