Why in the Ridge regression, the coefficients cannot be 0? In the second answer (https://stats.stackexchange.com/a/368426/287815) to the question (Why will ridge regression not shrink some coefficients to zero like lasso?), the OP found out that,
$β = /(^2+λ)$
My question:
Can't the $\beta$ be 0 if one of x or y is a zero vector?


what is the difference between "shrinking to exact 0" and "parameters can be 0"

If the OLS solution is non-zero (which means that $y$ is non-zero) then the ridge regression regularisation will not be able to shrink the parameters to exact 0.
As you found out the parameters of ridge regression can be zero when $y=0$ ($x=0$ makes no sense*). But that is only the case when $y=0$ and in that case the OLS solution is also zero (so it is not non-zero and there is nothi g to shrink). For this case it is not due to shrinking that the ridge regression parameters are zero, but due to the OLS solution being already zero.
BTW when you observe a continuous variable then the case of a zero $y$ vector and a zero coefficient will have zero probability.

* The vector $x$ won't be zero. Because that would be a useless model $y = \beta \cdot 0$.
