When we compute a Ridge regression model, do we need to compute the intercept separately from the slopes? As you know, the estimated $\beta$ for the ridge regression model is given by:
$\hat \beta = (X^TX - \lambda I_n)^{-1} X^T y$
But this formula is only for the slope parameters, excluding the intercept.
I'm just operating on pure intuition here, but can't we just append the constant column to $X$...
\begin{bmatrix} 1 & x_{11} & x_{12} & \ldots\\ 1 & x_{21} & x_{22} & \ldots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix}
...and add a diagonal element at the beginning of $\lambda I_n$ with the value 0?
\begin{bmatrix} 0 & 0 & 0 & \ldots\\ 0 & \lambda & 0 & \ldots \\ 0 & 0 & \lambda & \ldots\\ \vdots & \vdots & \vdots & \ddots \end{bmatrix}
