Suppose I have a system of equation $Y = VX$, where $V$ is a tall Vandermonde and full-rank:
$$V \triangleq\left(\begin{array}{ccccc} 1 & \lambda_1 & \lambda_1^2 & \lambda_1^3 & \ldots \\ 1 & \lambda_2 & \lambda_2^2 & \lambda_2^3 & \ldots \\ 1 & \lambda_3 & \lambda_3^2 & \lambda_3^3 & \ldots \\ \vdots & \vdots & \vdots & \vdots & \ddots \\ 1 & \lambda_r & \lambda_r^2 & \lambda_r^3 & \ldots \end{array}\right)$$
while $Y$ is a tall matrix while $X$ is a wide matrix. My goal is to find the vector containing all $\lambda_i$ with the only knowledge of $Y$. I know that $H_1=VV'$ and $H_2=V'V$ are both Hankel matrices. My first question is: there exist a factorization of $H_1$ or $H_2$ revealing the $\lambda_i$? I know about the Vandermonde factorization of Hankel matrices.
My second question is: does it help to solve $Y^\top Y = X^\top H_1 X$ or $YY^\top=VXX^\top V^\top$?
I was thinking if it can be associated to some sort of eigenvalue problem. I do not have matrix X in general.
On another side I am looking if something can be done also by means of Krylov subspaces, since $V = [1, \operatorname{Diag({\bf \lambda})}1, \operatorname{Diag({\bf \lambda})^2}1, \ldots]$.