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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!

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