# How to find the correspondence between ridge coefficients to modified ridge?

The ridge coefficients minimize a penalised residual sum of squares:

$$\hat{w}_{ridge}= \arg\min(\sum_{i=1}^{N}(y_{i}-w_{0}-\sum_{j=1}^{d}x_{ij}w_{j})^{2}+\lambda\sum_{j=1}^{d}w_{j}^{2}),\,\lambda\geq0$$

Now I want to show that the above minimisation is equivalent to:

$$\hat{w}^{c}=\arg\min(\sum_{i=1}^{N}(y_{i}-w_{0}^{c}-\sum_{j=1}^{d}(x_{ij}-\bar{x}_{j})w_{j}^{c})^{2}+\lambda\sum_{j=1}^{d}(w_{j}^{c})^{2}),\,\lambda\geq0$$

where $$\bar{x}_{j}$$ is the mean of the j-th element

what im trying to find is the Give the correspondence between $$w^{c}$$ and the original $$w$$

Would appreciate some guidance and help