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Typically, $X$ is $n\times p$, where $n$ is the number of samples and $p$ is the number of features. So, the shape of $\lambda I$ should be $p\times p$ in order for the expression to be mathematically correct. The identity term comes from the loss function of the Ridge Regression: $$L=||X\beta-y||^2+{\lambda}||\beta||^2 = (X\beta-y)^T(X\beta-y)+{\lambda}\... 0 No @whuber was not really recommending this approach. I would say the standard software approach, at least in R, is using offset. Ie if you want to regularise around b instead of 0, then add an offset of Xb, then your coefficients are for (Beta-b) See offset Param in https://www.rdocumentation.org/packages/lme4/versions/1.1-21/topics/lmer 1 If you use no regularization, i.e. alpha=0, they'll be the same/very similar when used the same random initialization, e.g. random_state=42. This is because Ridge class doesn't regularize the intercept term, but only the weights. When you include the intercept term, it becomes a member of your feature set and the column is assigned a weight, which is then ... 0 It says that U \in \mathbb{R}^{n \times p} so the columns are orthogonal but the matrix is not necessarily orthogonal in the sense that UU^T =I (\dim(UU^T) =p) 2 U has orthogonal columns but it's not square and isn't an orthogonal matrix. UU^T \neq I in general since it only has rank p and UU^T involves dot products of the rows which don't have to be orthogonal when p<n. If U was square then it would be the case that UU^T = I which makes sense since then \hat y = y since X is a bijection. I'm ... 4 From the context, I'm assuming that the \beta_j's are the regular least squares estimates, and the table is showing how they would be transformed under each of the listed methods. Best Subset: Because the columns are orthonormal, the least squares coefficients are simply \hat{B_j} = {x_j^{T}y}. (Orthogonality implies that they're given by \hat{B_j} = ... 1 Yes, you can obtain a rough estimate of the penalty parameter from standard OLS output. People in the old days used to compute$$ \lambda = \frac{ k \hat{\sigma }^2} {||\hat{\boldsymbol{\beta}}||^2},  where $k$ is the number of slope coefficients, $\hat{\sigma }$ is an estimated residual scale and $\hat{\boldsymbol{\beta}}$ is the vector of estimated ...