# Eigenvector of Hadamard matrix functions

Let $$X\in\mathbb{R}^{n\times n}$$ with SVD $$X=UDV^T$$. Are there known results regarding the eigenvectors of $$Y=X^{\odot g}$$? I am mainly interested in simple functions such as $$g(z)=z^2$$, i.e. $$Y_{ij}=X^2_{ij}$$.

I have tried finding it in Topics in Matrix Analysis without any luck so far.

Any help will be greatly appreciated!